STRUCTURES AND CRYSTAL CHEMHSTRY
OF
WOLLASTONITE AND PECTOLITE
by
CHARLES THOMPSON PREWITT
S.B., Massachusetts Institute of Technology
(1955)
S.M.,
Massachusetts Institute of Technology
(1960)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF
PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September, 1962
.
Signature of Author . . . . . . . . . . . . . . . .. ...
Department of Geology and Geophysics, August 20, 1962
Certified
by.
.
.-.-.-.
Thesis Supervisor
Accepted by..-.-Chairman, Departmental Committee
on Graduate Students
Structures and crystal chemistry of
wollastonite and pectolLte
by
Charles T. Prewitt
Submitted to the Department of Geology and Geophysics on
August 20, 1962, in partial fulfillment of the requirement for the
degree of Doctor of Philosophy.
ABSTRAC T
Wollastonite, CaSi.O3 , and pectolite, Ca 2NaHSi30 9 have
similar triclinic cells; because of this and their analogous chemical
compositions, they have been regarded as belonging to the same
mineral family. An examination of the results of least-squares'
refinement of three-dimensional x-ray diffraction data reveals that
although the structures contain similar metasilicate chains, .they are
not as closely isotypic as had been supposed. The principle differences are in the coordinations of calcium and sodium and in the way
these atoms share oxygens with the silicon atoms. The final R-factors
for wollastonite and pectolite are, respectively, 0.089 and 0.075.
The structures of both wollastonite and pectolite contain
pseudomonoclinic subcells which are joined together on (100) to give
an overall symmetry different from that of the subcell. The geometries of these sabcells and the possibility of their being joined in
different ways are important in explaining the types of twinning which
occur in wollastonite and pectolite. Twinning which produces an
overall crystal symmetry different from some subcell, known as
space group twinning, probably occurs in many types of silicate
minerals.
Programs written for the IBM 7090 digital computer, including
full-matrix crystallographic least-squares refinement and diffractometer settings programs, are included as part of a system designed
for automated crystal structure solution and refinement. Such a
system is not available at present but should be ready within two years.
Thesis Supervisor: Martin J. Buerger
Title: Institute Professor
-.--.
U-
,
Preface
This thesis is divided into three parts, A, B, and C. Part A
consists of four chapters, each of which is intended for publication.
Part B contains an account of work which was done during the thesis
period and which is not intended for publication in its present form.
There is, however, a possibility that some of this material will be
used as a base for future publication.
Part C contains four appen-
dices, the first three of which are descriptions and listings of
computer programs written during the period of thesis research.
It is hoped that these programs will be of use to 6thers doing crystal
structure work, and that they will eventually be incorporated into a
complete system for crystal structure investigations.
The fourth
appendix contains observed and calculated structure factor data on
wollastonite and pectolite.
Acknowledgements
The author gratefully acknowledges the advice and support
of Professor M. J. Buerger in this work on wollastonite and pectolite.
Crystals of wollastonite, pectolite, and larsenite were
kindly provided by Professor Clifford Frondel of Harvard University.
Mr. Alden Carpenter of Harvard loaned several specimens of
parawollastonite.
Pectolite and wollastonite were obtained from
Dr. George Switzer and Mr. Roy S. Clarke of the U. S. National
Museum.
The author would like to especially thank Professor A. Pabst
of the University of California for sending oscillation photographs
and specimens of Crestmore parawollastonite and for his comments
on parawollastonite.
Dr. Waldemar Schaller of the U.
S. Geological
Survey gave valuable advice on the pectolite-schizolite-serandite
series.
Many useful and informative discussions with Mr. Donald
Peacor have contributed greatly to the successful completion of
this work, particularly in connection with his investigations of
bustamite and rhodonite.
Mrs. Hilda Cid-Dresdner, Mr. Bernhardt
Wuensch, and Mr. Wayne Dollase are thanked for their interest and
help throughout the period of investigation.
Many conversations
with Dr. Charles Burnham about computing were especially useful.
Miss Gay Lorraine and Mrs. Shirley Veale contributed greatly
through their typing of the entire thesis.
Particular thanks are due
Miss Lorraine for proofreading and for her advice on the format of
the final draft.
My wife, Gretchen, helped greatly through her moral support
and through the preparation of many of the illustrations.
Finally, the support of the National Science Foundation
through part-time Research Ass istantships, and the M. I. T.
Computation Center through the use of the IBM 709 and 7090 computers
is gratefully acknowledged.
Table of Contents
Title page
i
Abstract
LL
Preface
tiL
Acknowledgements
iv
Table of Contents
v
List of Tables
x
List of Figures
xi
Part A
Chapter I.
pectolite
Crystal structures of wollastonite and
1
Attempts to solve the wollastonite structure from
Patterson projections
2
New diffractometer data
3
The three-dimensional Patterson function and its
solution
3
Refinement of the wollastonite structure
5
Attempt to fit pectolite into a wollastonite pattern
7
Preliminary refinement of the pectolite structure
8
Comparison of the pectolite and wollastonite structures 9
Chapter II. Comparison of the crystal structures of
wollastoniteand pectolite
14
Abstract
14
Introduction
15
Confirmation of the structure
16
Comparison of structures
18
Pseudomonoclinic symmetry
27
Distribution of Ca and Na
35
Interatomic distances and interbond angles
40
Twinning
48
Conclusions
56
Acknowledgements
57
----
7-
-
Vi
Chapter 111.
models
Drilling coordinates for crystal structure
58
Abstract
58
Introduction
59
Coordinate systems
59
Calculation of p and <
63
Discussion
66
Acknowledgements
67
References
68
Chapter IV.
Refinement of pectolite
69
Introduction
69
Selection of material
69
Unit cell data
71
Space group
73
Intensity collection
73
Refinement
75
Evaluation of the structure
78
Conclusions
87
References
88
Part B
Chapter V. Review of the crystallographic investigation of metasilicates
91
Status of structural work
93
Future possibilities for ino-metasilicate
investigations
95
References
Chapter VI.
Twinning in silicate crystallography
Structure and space group twinning
97
99
99
Implications of space group twinning
101
Energy relations in parawollastonite and bustamite
111
Space group twinning in other silicates
112
References
115
Chapter VII. Crystal structures related to wollastonite
and pectolite
116
Status of structural information
116
Pseudowollastonite
118
Hydrated calcium silicates
120
References
Chapter VIII.
Crystal structure of larsenite, PbZnSLO
Experimental work
122
124
124
Occurrence
124
Unit cell and space group
125
Intensity cellection
125
Solution of the structure
126
Refinement
131
Discussion of results
133
Oxygen packing
133
Substructures and heavy atoms
134
Direct methods
135
References
Chapter IX.
Automatic crystal structure analysis
136
137
Present capabilities
137
Future capabilities
138
Expected results of automated crystal structure
140
analysis
Part C Appendices
Appendix I. SFLSQ3, a structure factor and leastsquares refinement program for the IBM 7090
digital computer
Introduction
142
143
143
Common storage
145
Sense indicators
145
Tape usage
145
Punched output
150
Description of the main program and its subroutines 151
MAIN
151
READI
154
READ2
154
READ3
155
SET
155
LOOKUP
155
WEIGHT
155
SPGRP
156
REJECT
166
STAT
167
PRSF
167
STMAT
169
PRST
169
CHLNK3
169
TEST
172
CARD
173
DIST
173
Running the program
173
Data deck
173
Submitting a run
180
References
181
Program listing
183
Appendix II. Program for computation of drilling
coordinates for crystal structure models
209
Directions for running the program DRILL
210
Data cards for DRILL
210
Description of data deck
211
Description of output
212
Program listing
213
Appendix III.
Computation of diffractometer settings
215
Program description
215
DFSET
215
INDEXI
216
INDEX2
216
OUTPUT
217
MIFLIP
217
TERM
217
Speed of computation
217
Running the program
Data de ck
218
218
References
222
Program listing
223
Appendix IV. Observed and calculated structure
factors for wollastonite and pectolite
235
Wollastonite structure factors
235
Pectolite structure factors
264
Biographical note
290
List of Tables
Chapter I
1. Refined coordinates of atoms in wollastonite and
pectolite, all referred to pectolite origin
Chapter II
1. Unit cells of wollastonite, NaAsO 3 , and pectolite
2. Atom coordinates for wollastonite and pectolite
3. Atom coordinates of wollastonite and pectolite
referred to a pseudomonoclinic cell P2,/n
(origin
at 1)
4. Interatomic distances and Interbond angles in
wollastonite and pectolite
5. Coordination of oxygens in wollastonite and
pectolite
Chapter IV
1. Compositions of two pectolites and the hypothetical
composition computed from Ca 2NaHSi 0
2. Results of least-squares refinement ofpectolite cell
constants
3. Refined coordinates for pectolite
4. Interatomic distances in pectolite
5. Oxygen coordination in pectolite
Chapter V
1. List of crystallographically interesting metasilicates
10
17
23
33
41
45
72
74
79
82
86
92
Chapter VII
1. Classification of compounds with formula ABX3
117
Chapter
1.
2.
3.
127
128
132
VIII
Comparison of larsenite and forsterite cells
The forsterite structure. The larsenite structure.
Atom coordinates for larsenite
Appendix I
146
1. Symbols assigned to COMMON storage
2. Tape assignments for SFLSQ3 at M. I. T. Computation
Center
2
3. Derivatives of F c (hki) or F (hki) with respect to
-c
--152
varied parameters
4. Quanttties evaluated by general space group sub158
routine
17.
5. Expressions evaluated in CHLNK3
List of Figures
(The page numbers given are those of the figure legends.)
Chapter
1.
2.
3.
4.
5.
6.
7.
8.
9.
II
Projection of the wollastonite structure along b
19
Projection of the pectolite structure along b
21
Projections of the pectolite and wollastonite structures
along b
25
Projection of the wollastonite structure along c
28
Projection of pectolite along c corresponding to
that of wollastonite of Figure 4
30
Structure of wollastonite projected onto (101) of
four adjacent pseudomonoclinic cells
36
Structure of pectolite projected onto (101) of the
four pseudomonoclinic cells
38
Three ways in which the pseudomonoclinic unit can be
stacked on (100) to produce different overall diffraction
effects
49
Reciprocal lattices corresponding to the geometries
shown in Figures 8a, 8b, and 8c
52
Chapter III
1. Coordination group of atoms N 1 , N 2 ' N3 and central
atom N, and its relation to the spherical coordinate
system
2. Vectors used in the calculation of angle 43
61
64
Chapter IV
1. Pectolite structure projected onto (101)
80
Chapter VI
1. Schematic representations of the ways in which
pseudomonoclinic subcells can be joined to give
a different overall symmetry
2. Pseudomonoclinic CaSi.O3 subcells joined on (100)
to give wollastonite (top) and parawollastonite (bottom)
3. Schematic representations of the wollastonite (top)
and bustamite (bottom) structures
4. Result of joining CaSiO subcells as described in
Figure 3 to give the woAastonite structure (top)
and the bustamite structure (bottom)
102
104
107
109
Chapter VIII
1.
Pb
1 +Pb2M 8 (xy 4 )
map of larsentte
129
-~.----4 U
XIl
Part A
Chapter I.
The Crystal Structures of Wollastonite and Pectolite.
Wollastonite, CaSi.O3 , and pectolite, Ca 2NaHSi 309, belong
to the same mineral family.
They have the same symmetry, P1,
and similar cells:
Wollastonite
Pectolite
a
7.94
7.99
b
7.32
7.04
c
7.07
7.02
a
90 021
900311
p
95 22t
95 11i
y
103026t
102028!
Z
6CaSiO3
A
2Ca 2NaHSi3 09
The structure of pectolite was solved by Buerger
2, and the
structure of wollastonite was solved shortly thereafter by Mamedov
3
and Below .
Curiously enough, although the two structures have
Buerger, M. J. The arrangement of atoms in the wollastonite
group of metasilicates. Proc.Nat.Acad.Sci., 42, (f95-6) 113-116.
2 Buerger, M. J. The determination of the crystal structure of
pectolite, Ca 2NaHSi3 0 9. Zeit. Krist,, 108, (1956) 248-261.
233
Mamedov, K. S. , and N.V. Belov. Crystal structure of wollastonite.
Doklady Akad. Nauk. SSSR, 107 (1956) 463-466.
rather similar arrangements,
they are not the same, and they are
described by different sets of coordinates for corresponding atoms.
For two crystals of the same family to have different
coordinates suggests that one of the structures might be incorrect.
Since pectolite was solved in this laboratory, we were prejudiced
in favor of the correctness of that structure.
Accordingly we
attempted to fit the wollastonite structure to the pectolite pattern
with curious consequences which are noted below.
Attempts to Solve the Wollastonite Structure from
Patterson Projections.
The structure of pectolite had been solved from the Patterson
projections P(xy), P(yz), and P(xz).
These were based upon
intensities determined from precession photographs.
The intensities
were rather crude since a coarse crystal had been used, and the
intensities had not been corrected for absorption.
In spite of these
poor data, the structure was readily solved by constructing minimum functions based upon these three Patterson projections.
The same strategy, when applied to wollastonite, was not
successful.
Patterson peaks corresponding to the Ca1+2
inversion peaks in pectolite were used for image poitts, and they
gave pectolite-like projections.
But the structures corresponding
to these projections had excessive distances from Si.3 to some of
the oxygen atoms of its tetrahedron, and their structure factors had
high values of R.
This failure was attributed to poor data, so that,
in due course, a new set of intensities was measured with the aid
of a single-crystal diffractometer.
New Diffractometer Data
The copper hemisphere for wollastonite contains 1, 769
reflections.
The intensities of all of these which were within
recording range of our instrument (namely 1, 503) were measured
with the aid of a special diffractometer
detector.
using a Geiger tube as
Due care was exercised not to exceed the linearity limit
of the detector.
The resulting data were corrected for Lorentz
and polarization factors, and then an approximate absorption
correction was applied
2
The Three -D imens ional Patterson Function and Its Solution.
The resulting F(hkl)ts were used to compute a threedimensional Patterson function.
This was studied in sections
Buerger, M. J. , and N. Niizeki. The correction for absorption
for rod-shaped single crystals. Amer. Mineral. 43 (1958)726-731.
2 Liebau, Friedrich. Uber die Struktur des Schizoliths. Neues Jb.
Mineral. , (1958) 227-229.
U
parallel to (010) made at intervals of b/30.
Although this Patterson function did not have a clear-cut
indication of the obvious image point to be used to start the minimum-function routine, as pectolite,had, a peak was present in a
location corresponding to that of a Ca +
pectolite.
inversion peak in
This was used in forming an M 2 (xyz) function.
This
preliminary function yielded an approximation to the substructure
This revealed the
which closely resembled that of pectolite.
location of Ca 3 (corresponding to Na of pectolite) and this atom
location permitted construction of another .M2 (xyz) function.
With
these two M 2 (xyz) functions, a stronger M(xyz) function was
formed.
This still resembled pectolite,
except that appropriate
locations for 07 and 08 could not be found, and an ambiguity
existed in placing Si , Si.2 and 0
.
None of the structures based
upon this solution of the Patterson function had a small value of R,
and none could be refined below R = 26%.
Since a pectolite-like model of wollastonite had evidently
failed, the model proposed by Mamadov and Belov2 was tested.
Buerger, M.J. The determination of the crystal structure of
pectolite, Ca 2NaHSi309. o_. cit.
2 Mamedov, K. S.,
and N. V.
Belov, op. cit.
Using our new values of F(hkl) and Mamedov and Belovis
coordinates,
the initial value of R was 31%, and this proved to be
refinable, as noted later.
It seemed desirable to re-examine the Patterson function in
the light of these new coordinates.
Accordingly, they were used as
a basis for finding the Ca1+2 inversion vector, and this was used
to decompose the Patterson function.
As in the earlier trial, this
permitted forming an M 2 (xyz) function, then an MA(xyz) function
based upon all the calcium atoms.
The resulting peaks of the final
minimum function located all the atoms of the model on which the
original inversion peak was based, and with correct weights.
It
was therefore evident that this solution of the Patterson corresponds
to the actual structure of wollastonite.
But it is also true that, as
discussed in the first part of this section, if a pectolite-like image
point is used as a start, a pectolite-like structure is predicted.
This curious situation made it advisable to check the correctness
of the pectolite structure.
(The confirmation and preliminary
refinement of the pectolite structure are reported in a subsequent
section of this paper.)
Refinement of the Wollastonite Structure
Mamedov and Belov based their solution of the wollastonite
structure on finding sign relations among 204 hOl reflections. This
provided signs for 30% of the reflections and permitted computing a
crude electron-density map.
With the aid of the peaks of this map,
and on the basis of experience with cuspidine and xonotlite, the
Patterson projection was interpreted.
R = 24% for 224 reflections.
This gave a model for which
Further refinement of this important
structure is obviously called for.
We started with our extensive Geiger-counter data for
1, 503 F(hkl) t s and Mamedov and Belov t s original coordinates
(except that they were transformed to the pectolite origin by adding
We employed Busing and Levy t s least-squares refi nement
T0).
program
for the IBM 704 computer, using an appropriate weighting
scheme.
After the initial cycle to find an approximate scale facbr,
the value of R started at 31%.
After four cycles of refinement using reasonable fixed
isotropic
temperature factors, the value of R was reduced to 11.4%.
The isotropic temperature factors (as well as coordinates) were
allowed to vary in three succeeding cycles, and this reduced R to
9. 3%.
An additional cycles in which the anisotropic temperature
factors were refined reduced R to 8. 9%.
The final coordinates of
the atoms, their standard deviations, and their isotropic temperature factors are listed in Table 1.
Busing, W.R., and H.A.Levy. A crystallographic least-squares
program for the IBM 704. (Oak Ridge National Laboratories,
Oak Ridge, Tennessee, 1959).
Attempt to Fit Pectolite into a Wollastonite Pattern.
Wollastonite and pectolite have long been regarded as
members of the same family.
Yet the structure proposed by
Mamedov and Belov for wollastonite and that proposed by Buerger
for pectolite (and later by Liebau
for schizolite) are different.
Since we confirmed the correctness of the wollastonite structure,
there arose a suspicion that the structure proposed for pectolite
was incorrect, and that it perhaps had the wollastonite structure.
From the earlier investigation2 there were available the
2
following sets of reduced intensity data: F (hk0),
2
F (hOl),
and
In addition, the Patterson projections based upon these
F 2(Okl).
sets, of F
2tt
s were available,
V
namely P(x), P(xz) and P(_yz).
One of the fundamental differences between the wollastonite
structure and Buerger t s pectolite structure is the location of Ca 1
and Ca
.
These atoms were used in the original pectolite investi-
gation for the initial M
-2
solve the structure.
minimum functions which were used to
The locations were found by making use of the
conjugate-peak relation.
Knowing the general location of Ca
and Ca2 in wollastonite,
Liebau, Friedrich. Uber die Struktur des &chizoliths. Neues Jb.
Mineral., (1958) 227-229.
2 Buerger, M. J. The determination of the crystal structure of
O. Cit.
pectolLte, Ca 2NaHSi.3 0;
a peak was found in the Patterson projection P(xz) of pectolite
whose location would correspond to an inversion peak for Ca
and
Ca2 if these atoms had the locations they have in wollastonite.
A
minimum function M2(xz) was found using this peak as an image
point, and interestingly enough, a wollastonite-like pattern of peaks
was revealed.
The other two projections of the structure, namely
M 2(xy) and M 2 (yz), were formed from the Patterson projections
P(xy) and P(yz).
Each had a wollastonite-like pattern.
These projections provided the approximate values of the
three coordinates xyz for each atom.
But this structure turned
out to have a discrepancy R(hkl) = 62% based upon the reflections
F(hkO), F(h1), and F(Ok).
An attempt
was made to refine these
three sets together by least-squares as described later, but R(hkl)
could not be reduced below 52%.
In spite of the possibility of
forming minimum functions having a wollastonite-like pattern of
atoms, such a structure cannot be refined and must be incorrect.
Preliminary Refinement of the Pectolite Structure.
Since pectolite evidently does not have a wollastonite-like
pattern of atoms, it appeared that the solution of the pectolite
structure given by Buerger
was probably correct.
The original
Buerger, M. J. The determination of the crystal structure of
pectolite, Ca 2NaHSi3 0 9; op. cit.
9.
structure determination included a modest refinement by difference
maps which left the discrepancies at
R(hk0) = 16%;
R(Okl) = 21%; R(hOl)
= 25%.
To improve this state of the structure, the original sets of F(hkO),
F(Okl),
and F(hOl) were treated together as a single three-
dimensional least-squares refinement with a different scale factor
applied to each set, using the Busing and Levy program for the IBM
704 computer .
In two cycles the refinement converged to a set of
coordinates for whihh the over-all R(hkl) was 17%.
Considering
the crude nature of the original intensity measurements, this can
be regarded as a successful refinement, and it confirms the
correctness of the original pectolite structure determination.
The
new values of the coordinates are given in Table 1.
Comparison of the Pectolite and Wollastonite Structures
As a result of this investigation it is evident that wollastonite and pectolite, though belonging to the same mineral family,
have distinct but related structures.
The relations between the
two structures will be discussed in a forthcoming paper.
Busing, W.R.,
and H.A. Levy, op.cit.
Table
I
Refined Coordinates of Atoms in Wollastonite (upper values) and Pectolite (lower values), All
Referred to Pectolite Origin
Ca 1
(Ca1 1 )
x
U-(x)
y
U(Y)
z
(z)
B
0.1985
0.0001
0.4228
0.0001
0.7608
0.0001
0.41
0.0001
0.45
0.0001
0.37
0.0002
0.24
0.404
0.143
Ca
2
(Ca11 )
0.2027
0.0001
0.157
Ca
3
(Ca)
Na
Si
0.4966
(SL 1 r)
0.1852
0.221
0.0001
0.0001
0.2495
0.0001
0.3870
0.402
0.4720
0.656
0.735
0.0002
0.7640
0.861
0.916
0.448
1
0.9293
0-.854
0.0002
0.2687
0.337
Belovis designations for atoms of wollastonite in parentheses. Because of the poorer intensity
data for pectolite, standard deviations of coordinates and individual temperature factors are
omitted.
(Table I continued)
Si
2
(Si)
x
c(x)
y
0.1849
0.0002
0.9545
0.210
3
S
11)
0.3970
0.0002
(01)
0.4008
0.0005
0.3037
0.0005
0.3017
0.171
0.2314
0.7259
0.0005
0.4635
0.0006
0.9374
0.839
B
0.0002
0.24
0.0560
0.0002
0.22
0.8019
0.0005
0.48
0.0005
0.37
0.0006
0.60
0.0006
0.64
0.875
0.0006
0.8302
0.943
0.0006
0.4641
0.538
0.496
0.0005
0.2692
(z)
0.148
0.702
0.185
04
0.0002
0.212
0.322
03
0.7235
z
0.344
0.735
0.348
02
0.0002
0.945
0.451
0.4291
o-(y)
0.0006
0.4655
0.541
(Table I continued)
05
(Omi1!)
x
cY(x)
y
c(y)
z
c(z)
B
0.0154
0.0005
0.6254
0.0006
0.7343
0.0006
0.63
0.0006
0.71
0.0005
0.37
0.0005
0.51
0.0005
0.68
0.070
06
(0
)
0.0175
0.393
0.0005
0.053
07
(O IV)
0.2732
(0
)
0.2713
0.0004
()0
0.2188
0.260
0.5118
0.0005
0.8717
0.0005
0.1784
0.182
0.0919
0.275
0.0005
0.906
0.0005
0.7353
0.179
0.533
0.402
0
0.0006
0.896
0.396
08
0.1319
0.171
0.0940
0.275
0.0005
0.2228
0.381
13This work was supported by a grant from the National Science
Foundation.
Part of the computation work was carried out at the
M. I. T. Computation Center.
iV
id
Chapter II.
Comparison of the Crystal Structures of Wollastonite
and Pectolite.
Abstract
Wollastonite,
CaSiO 3 , and pectolite,
CaNaHSi3 0
have
similar triclinic cells; because of this and their analogous chemical
compositions they have long been regarded as belonging to thessame
mineral family.
An examination of the results of least-squares
refinement reveals that although the structures contain siinilar
metasilicate chains,
supposed.
they are not as closely isotypic as had been
The principal differences are in the location of the
large cations between layers of oxygen atoms parallel to (101) and
in the relative orientations of the metasilicate chains.
The structures of both wollastonite and pectolite contain
pseudomonoclinic subcells which are joined together on (100) to
give an overall symmetry different from that of the subcell.
The
geometries of these subcells and the possibility of their being
joined in different ways are important in explaining the types of
twinning which occur in wollastonite and pectolite.
1.5
Introduction
PectolLte and wollastonite belong to the same mineral
family and have distinct but related crystal structures.
and Prewitt (1962)
by Buerger (1956)
2
Buerger
have shown that the crystal structures proposed
for pectolite and by Mamedov and Belov (1956)
for wollastonite are correct.
3
The present paper describes in
detail the similarities and differences between the two structures
and attempts to explain some of the puzzling features of these
minerals which have been observed in the past.
Buerger, M.J., and C. T.Prewitt (1962).
of wollastonite and pectolite.
2 Buerger, M.J. (1956).
The crystal structures
Chapter I, this thesis.
The determination of the crystal structure
of pectolite, Ca 2NaHSL3 0 9. Z. Krist., 108, 248-261.
233
3 Mamedov, K.S.,
Crystal structure of
and N.V. Belov (1956).
wollastonite.
Doklady Akad. Nauk SSSR,
107, 463-466.
16
Confirmation of the Structures
Unit cell information for pectolite, wollastonite,
NaAsO3 is given in Table 1.
and
The similarity of the triclinic cells
(space group PI) and the analogous chemical compositions indicate
that these substances might have the same structure.
A comparison
of the results given by Mamedov and Beloy (1956) 1 for wollastonite
and Liebau (1956)2 for NaAsO3 shows that these two structures
are identical.
If either of these is compared to the pectolite
structure given by Buerger (1956)
3
certain discrepancies are noted.
For example, while in both structures the silicate chains have
repeat units of three tetrahedra, they do not have the same orientation in the two structures, and the large cations all lie in a sheet
parallel to (101),
but do not have the same distribution in the
sheet.
In order to determine whether these differences were real,
1 Mamedov, K. S.,
and N. V. Belov, op. cit.
2 Liebau, Friedrich (1956).
natrlumpolyarsenats,
Uber die kristallstruktur des
(NaAsO )x.
3 B,
Buerger, M. J., of. cit.
Acta Cryst.,
9, 811-817.
I
Table 1.
Unit Cells of Wollastonite, NaAsO
Wollastonite
Ca3Si 30
7. 94
7.32A
7.07
Pectolite
Na 3As30
A
A
and Pectolite.
Ca 2NaHSi3 0
8.07
7. 99
7.44
7.04
7. 32
A
7.02
90082!
90 0
90 031
95 221
91 0 30t
950 11?
103 26t
104 0
102028t
18
Buerger and Prewitt (1962)
collected three-dimensional x-ray
intensities for wollastonite and refined the structure by leastsquares.
In addition, they refined pectolite using relatively
2
crude intensity data originally obtained by Buerger (1956)2.
The
final R factors for wollastonite and pectolite, respectively, were
8. 9% and 17%.
When the structures and x-ray data were inter-
changed, the wollastonite data could not be refined below 26% and
the pectolite data not below 52%.
Therefore,
the structures as
proposed must be correct but different.
Comparison of Structures
Figures 1 and 2 are projections of the wollastonite and
pectolite structures along b, respectively.
Coordinates for the
figures in this paper were given originally by Buerger and Prewitt
(1962)
and are reproduced in Table 2 with a few changes so that
all coordinates refer to atoms in or near the same silicate chain.
Unless otherwise noted, the origin for the figures is that of the
coordinates of Table 2.
2
Buerger,
M. J., and C. T. Prewitt, oR. cit.
Buerger,
M. J.,
op. citt.
r
10
FLgure A.
Projection of the wollastonite structure along b.
10UIS 0
-90,9
Figure 2.
Projection of the pectolite structure along b.
the Si
Although
and Si2 tetrahedra should not be exactly super-
imposed, they are presented this way to simplify the
drawing.
Au\S D
Pulso
Table 2.
Atom Coordinates for Wollastonite and Pectolite.
Wollastonite
Atom
Pectolite
x
y
z
Ca 1
.1985
.4228
.7608
Ca
.2027
.9293
.5034
Atom
x
y
z
Ca 1
.857
.596
.146
.7640
Ca
.843
.074
.139
.7505
.5280
Na
.552
.265
.344
. 1852
. 3870
. 2687
Si
. 221
.402
. 337
Si 2
.1849
. 9545
. 2692
Si.
. 210
. 954
.344
Si.
.3970
. 7235
. 0560
Si
.451
. 735
. 148
01
.5709
.7686
.1981
01
.652
.788
.125
02
.4008
.7259
02
.322
.702
-. 057
0 3
. 3037
.4635
.4641
0 3
.185
.496
.538
04
.3017
. 9374
.4655
04
. 171
. 839
. 541
-. 0154
.3746
.2657
.070
.393
.171
06
-. 0175
.8681
.2647
06
. 053
.896
.179
07
.2732
.5118
.0919
07
.396
.533
.275
08
.2713
.8717
.0940
08
.402
.906
.275
0
. 2188
. 1784
.2228
0
. 260
. 182
.381
2
Ca
3
05
3
-.
1698
2
1
2
3
05
Figures 1 and 2 reveal that the structures are topologically
different.
Si
For example, the bonds between the superposed Si
tetrahedra and the Si3 tetrahedron are reversed.
and
The large
cations, however, occupy approximately the same positions in the
two structures.
One immediately wonders whether the misfit is
merely a result of identical structures being oriented in different
ways.
The authors spent some time in considering this possibility
and found that the structures can be oriented so that the silicate
chains are nearly superposed.
This is accomplished by inter-
changing the a and c axes and shifting the origin of one of the
structures.
Figure 3 shows projections of wollastonite and
pectolite along b in which the silicate chains are oriented to
correspond with one another.
The y coordinates of the atoms in
the respective chains are about as close to each other as are the x
and z coordinates.
are not similar.
The locations of the large cations, however,
In fact, the large cations occupy completely
different interstices between oxygens when the structures are
oriented in this way.
The structures must then be different even
though the packing of oxygens is roughly similar.
The authors
originally thought that this kind of comparison might be more
useful than the conventional one, but further investigation as
reported below showed that the conventional orientations should
be retained.
Figure 3.
Projections of the pectolite and wollastonite structures
along b.
The origin and orientation of wollastonite has
been changed from that of Fig. I so that the silicate
chains in wollastonite have the same orientation as in
pectolite.
. a sin7
-c sina
asiny
c sina
PEC TOLIT E
WOLLASTONITE
?,7
Pseudomonoclinic Symmetry
A number of investigators have commented on the unusual
features which are observed in x-ray diffraction photographs of
wollastonite and pectolite.
Among these features are a mirror
symmetry between relative even-numbered reciprocal lattice levels
normal to b, and the presence of a substructure along b with a period
of b/2 as indicated by the average spot intensity in even-numbered
reciprocal lattice levels normal to b being greater than in the oddnumbered levels.
Ito (1950)
discussed these effects and proposed
that triclinic wollastonite could be constructed by starting with a
hypothetical monoclinic cell and shifting successive cells along (100)
in increments of *b.
To this pseudomonoclinic cell he attributed a
symmetry of P 2/n or P 21 /m.
Figures 4 and 5 show that both
wollastonite and pectolite can indeed be so regarded, and that the
space group of this unit, which is infinite in the b and c directions
and one unit cell thick along a, is P2 1 /m.
Liebau (1956)2 also
noticed that NaAsO3 can be so regarded.
This shift between successive pseudomonoclinic units not only
determines the triclinic symmetry in pectolite and wollastonite, but
is also the key to twinning in these minerals and probably will be
found to be significant in other triclinic metasilicates as well.
Ito, T. (1950). X-ray studies on polymorphism. Maruzen, Tokyo,
93-110.
Liebau, Friedrich, op. cit.
Figure 4.
Projection of the wollastonite structure along c.
The mirror planes and screw axes are elements
of the pseudomonoclinic unit.
g'u!S D
mm
-~
Figure 5.
Projection of pectolite along c corresponding to
that of wollastonite of Fig. 4.
Note that the
orientations of the silicate chains are different In
the two structures.
'uIso
The results of the different shifts which can be made are discussed
in another section of this paper.
It is interesting to note how closely the refined atom
coordinates conform to the monoclinic symmetry.
The coordinates
were refined in the space group P 1 so that all atoms were in the
general position with no restrictions on x, y or z.
Table 3 gives
the wollastonite and pectolite coordinates transformed to the
pseudomonoclinic cell using the cell transformation'
Pseudomonoclinic
1
Triclinic
-
0
0
1
0
0
0
1
and adding 0.125 to the transformed y coordinates.
For the atoms Ca3' Si 3
O , 02 and 0
which would be on the mirror plane in P 2 1 /m,
should be . 250 or . 750.
in wollastonite,
the y coordinates
The very small deviations of about
.
001
(or .007A) are within the expected accuracy of the measurements.
The other atoms which are related by the mirror planes also show
little deviation from the positions they would occupy if the monoThis transformation is not exact when there is a departure of a
(triclinic) from 900; a in wollastonite is nearly 900 and
departs from 900 by only 31
in pectolite.
Table 3.
Atom Coordinates of Wollastonite and Pectolite referred
to a Pseudomonoclinic Cell P2 1 /
Pseudoequipoint
Wollastonite
Atom
x
(Origin at i).
Pectolite
y
z
Atom
x
y
z
2Ca1
. 1.985
. 4982
. 7608
2Ca1
. 857
2Ca
. 2027
. 0038
. 7440
2Ca
. 843
2Ca 3
. 5034
. 7496
. 5280
2Na
. 552
. 252
. 344
2Si1
. 1852
.4657
. 2687
2Si 1
. 221
.472
. 337
2Si 2
. 1849
. 0333
. 2692
2Si
. 210
1.026
. 344
2c
2Si
2i3
.3970
.7492
.0560
2Si
.451
.747
.148
2e
201
.5709
.7509
.1981
201
.652
.750
.125
2e
20 2
.4008
.7507
1698
20 2
.322
.746
.057
4f
20 3
.3037
.5126
.4641
203
.185
.575
. 538
204
. 3017
. 4870
.4655
20 4
. 171
. 921
. 541
2
-.
2
2
-.
. 507
.146
002
.139
205
-.
0039
.5035
.2657
20 5
.070
.500
.175
206
-.
0044
.9975
.2647
206
.053
1.008
.179
207
.2732
.5685
.0919
20 7
.396
.559
.275
208
.2713
.9287
.0940
208
.402
.930
.275
20 9
. 2188
. 2487
.2228
20 9
. 260
. 242
.381
clinic symmetry held.
If the space group P 2k/n is assumed, then
the following paired atoms are equivalent and occupy the general
position:
Ca
Ca3'
Si 3
1
, Ca2
O,
Si'
02 and 0
S 2
03,
05, 06
07, 08*
are each in the special positions m.
Although the atoms in pectolite have the same pseudoequipoint distributions as do those in wollastonite, the refined
coordinates for the pectolite atoms do not conform as closely to the
monoclinic symmetry.
For example, the y coordinate of 0
.242 as compared with .250 if 0
is a deviation of about 0.06A.
were on the mirror plane.
is
This
A detailed refi nement of pectolite
now being carried out in this laboratory will determine whether
these deviations in the pectolite coordinates are real.
The pseudosymmetry gives rise to the strong substructure
along b, since all the atoms except 07 and 08 in both wollastonite
and pectolite and Na in pectolite have x and z coordinates similar
to another atom of the same type located -b away.
It is interesting
that Ca3 and its inversion equivalent in wollastonite have similar
x and z coordinates and are approximately Lb apart, while Na and
its inversion equivalent in pectolite do not have similar x and z
coordinates even though they are separated by -b.
This is one of
the major differences between these minerals.
The similarities and differences between wollastonite and
pectolite brought out in this section seem to be more useful in
describing the structures than do those in the last section where a
special orientation of one of the structures was used to show how
the silicate chains could be superposed.
Because of this, the
authors feel that the conventional orientation should be retained
and that while the other aspect is interesting, it will not be useful
except, for example, in a classification of metasilicates based on
oxygen packing.
Distribution of Ca and Na
The main difference in the wollastonite and pectolite
structures is in the way the large cations are distributed in the
sheet parallel to (101).
Figures 6 and 7 show parts of the
wollastonite and pectolite structures projected onto (101) of the
pseudomonoclinic cell.
Only the silicate chains on the upper
side of (101) are shown.
The Ca in wollastonite are arranged in a nearly planar
hexagonal pattern which is separated into bands 3 Ca columns
wide running in the b direction and parallel to the silicate chains.
Fig. 6
Structure of wollastonite projected onto (101) of
four adjacent pseudomonoclinic cells.
Only the
silicate chains on the near side of (101) are
shown.
The axial directions are for the pseudo-
monoclinic cell.
[ol
38
Figure 7
Structure of pectolite projected onto (101) of the
four adjacent pseudomonoclinic cells.
Only the
silicate chains on the near side of (101) are
shown.
The most likely location for the hydrogen
bond is indicated by the dotted line between the
labeled 03 and 0
.
The Na and Ca in pectolite are also arranged in a hexagonal pattern
with bands consisting of 2 Ca columns bounded on either side by a
half-filled Na column.
by
Because each of these bands is translated
1b with respect to neighboring bands, every other Na in a
particular Na column is missing.
This is necessary because
otherwise the Na-Na distances between neighboring Na columns
would be much too small.
Interatomic Distances and Interbond Angles.
Interatomic distances and angles given in Table 4 were
calculated for wollastonite and pectolite using a computer program
written by Busing and Levy (1959) .
The Si-O distances in both
wollastonite and pectolite show a considerable variation,
the
lengths apparently being determined by the coordinations of the
oxygens.
Table 5 gives the coordination and distances of cations
around the oxygens.
It can be seen that,
for O
through 06 of
wollastonite, the Si-O distances are longer when an oxygen is
coordinated by 3 Ca than when coordinated by 2 Ca.
Busing, William R.,
and Henri A. Levy (1959).
function and error program for the IBM 704/
Laboratory Central Files No. 59-12-3.
The situation
A crystallographic
Oak Ridge National
Oak Ridge, Tenn.
1-95.
Table 4.
Interatomic Distances and Interbond Angles in Wollastonite
and Pectolite.
Interatomic Distances
Atoms
Si 1
WollastonLte
03
1.61
05
1. 572
1.59
07
1.659
1.60
09
1.647
1.67
1.62
1.62
O4
06
1.617
1.63
1.58 1
1.61
08
1.650
1.75
09
1.637
1.58
0
1.62
1.64
av
Si2
Si3
Ca
A
Pectolite
av
0
1.59
1. 63
1.59
02
07
1.59
1.68
1.665
1.68
08
av
1.673
1.63
1.63
1.65
0
2.30 2
2.33
0
03
2.272
2.31
2.324
2.35
05
2.30 2
2.35
O r
5.
-
2.44
06
07
2.272
0
2. 31
av
2.34
2.412
4
2.35
A
(Table 4 continued)
Atoms
Ca
Ca
2
3
Wollastonite
Pectolite
01
2.501
2.30
02
04
2.421
2. 31.6
2.25
05
2.368
2.51
06
06
2.316
2.35
0
2.406
.8
av
Na
-
2.45
6
2.38
0
2.43
02
2.34
03
2.42
03
04
2.335
2.44
0 41
2.34
09
2.642
2.390 (excluding
Ca 3 -0 9)
Oav
2.32
2.36
02
2.32
0
2.46
0
2.54
07
2.50
0!1
2.97
08
2.57
08
.
2.97
09
2.32
0
2. 45 (excluding
Na-071
Na-0 8)
av
3
~+
(Table 4 continued)
Wollastonite
Atoms
1
Si
3.165
3.10
Si.
3
3.11 6
3.04
Si 3
3. 125
3.11
Si
Si
2
2
Si - 0 - Si
1
9
2
Si
Si
2
3
- 0
-0O
Pe ctolite
I
I
Interbond Angles
149. 00
149. 50
8
- Si.
3
140.0
134.70
7
- Si.
1
139.00
136.10
is different for 07 through 0
because these oxygens are coordinated
by 2 Si as well as Ca, showing that Paulingts rule does not hold.
The Si-O distances here are larger than in either of the examples
above.
The coordination of oxygens in wollastonite can thus be
divided into three classes:
1.
Oxygen coordinated by one Si and three Ca (01,
02'
3' 0 4);
2.
Oxygen coordinated by one Si and two Ca (0,Y 06)
3.
Oxygen coordinated by two Si and one
Ca (07, 08'
0 ).
It is felt that an analysis of the pectolite interatomic
distances before a detailed refinement is completed would be
premature, particularly since the range in Si-O distances is so
great, varying from 1.58A for Si2- O
to 1.75X for Si2 08'
This may be a real variation, however,
eince Morimoto, Appleman
and Evans (1960)
reported a range of 1. 58 - i.74AR for Si-O in
clinoenstatite.
Ca
and Ca2 in wollastonite are octahedrally coordinated
by six oxygen atoms at average distances of 2. 383
2. 388
A
for Ca2.
and
Ca3 is surrounded by six oxygen atoms at an
average distance of 2. 39 0
1
for Ca
A
and by an additional oxygen (09) at
6
Mor imoto, N. , D.E. Appleman and H. T. Evans (19 0). The
crystal structures of clinoenstatifte and pigeonite. Z. Krist., 114,
120-147.
Table 5.
Coordination of Oxygens in Wollastonite and Pectolite.
Wollastonite
Atoms
0
Pectolite
Interatomic
Atoms
Distances
Si3
Ca 1
1.. 59
9
2.30 2
Ca
2.316
2
Ca
3
Si.
3
Ca
1
Ca
Ca
03
I
2
3
1. 59 9
02
05
2.30
2
Si
3
1.68
2.31
2. 36 9
Ca
2.25
2. 34 9
Na
2.32
Si
1.63
2.42 9
Ca3
2.335
Si 2
1.617
Ca 2
Ca
03
2
1
Ca
1
2.35
2.46
1.63
2.42
Si2
Ca
2.44
Na
2.54
Si
1.59
Ca 3
2.349
Si
1.572
1
Ca
Ca 1
Ca 3
3
2.33
2. 272
2. 324
04
1.59
Si3
a1
2.439
1.618
SLi
0
Interatomic
Distances
0
2
05
1
2.32
2.35
Ca 1
2.302
Ca
Ca
2.50
Ca 1
2.44
Ca
2.51
2
1
2
(Table 5 continued)
06
O
Si2
1.58
Ca 1
Ca 2
06
S2
1.61
2.272
Ca
2.34
2.50
Ca2
2.35
Ca 2
2.45
Si.
1.60
Si
1.65
Si 3
1.665
S
3
1.68
Ca 1
2.412
Na
2.50
07
Na t
o8
Si.
2
Si 3
Ca
2
1.65
33
0
0
8
1.67
2.40 6
Si 21.75
2
Si
3
Na
1.63
2.57
Nar
0
9
Si.
1
Si.2
1.64
Ca 3
2.642
7
1.637
0
9
Si.
1
Si 2
1.67
0
2.39
1.58
=====;PW
4+7
2.64
.
.
The Ca- 0
sheet is similar to a brucite sheet separated
into slabs running along b and distorted by the presence of 0 .
The coordination of Ca
that of Ca
of 2. 35.
and Ca
and Ca
in pectolite is similar to
in wollastonite with an average Ca - 0 distance
and an average Ca 2- 0 distance of 2. 36X.
Na is
surrounded by six oxygen atoms at an average distance of 2.45AX,
and two further oxygen atoms, both at 2. 97A.
The most probable location for the hydrogan bond in pectolite
is between 03 and 0 .
These atoms are separated by 2.44A which
is short even for a hydrogen bond.
This is represented by a dotted
line between the labelled 03 and 0
in Fig. 7.
This separation is
much less than between similar atoms in the wollastonite chain as
seen in Fig. 6.
Buerger (1956)
attributed the presence of a
shorter b axis in pectolite than in wollastonite to the hydrogen bond.
In both structures, the silicon-oxygen-silicon angles have the same
distribution, with Si - 0 -Si2 being larger than Si2-0 8
Si
3
-
0
7
-Si,
sh
which are approximately equal.
3 and
This difference results
from the somewhat abnormal location of 0 9in the oxygen sheet.
Buerger,
M.J.,
o.
cit.
Twinning
One of the most interesting aspects of the pectolite and
wollastonite structures is the way in which twinning can occur.
Mention has already been made of the way in which pseudomonoclinic
subcells are fitted together to give a triclinic structure.
shown schematically in Fig. 8b.
This is
It is also possible to reverse the
shift between successive pseudomonoclinic subcells to give the
sequence shown in Fig. 8a, which corresponds to a twinned
structure.
If, however, the shift is reversed in each successive
cell, the effect is as shown in Fig. 8c. This sequence was suggested
1
2
by Ito (1950)
for parawollastonite which was originally
distinguished bfrom wollastonite by Peacock (1935a)
3
Peacock (1935b)4 found that the twin law in pectolite is such
that b is the twin axis with (100) as the composition plane.
consistent with Fig. 8a since a rotation of 180
This is
around b of one of
the pseudomonoclinic units would be approximately equivalent to
1 Ito,
T.,
o . cit.
2 The Commission on Mineral Data recommended at the meeting
of
the International Mineralogical Association in Washington, D. C.,
April 17-20, 1962, that wollastonite and parawollastonite be redesignated wollastonite-T
Peacock, M.A. (1935a).
Am. J. Sci.,
and wollastonite-2M, respectively.
On wollastonite and parawollastonite.
30., 495-529.
Peacock, M.A. (1935b). On pectolite.
Zeit. Krist. 90,
97-111.
Figure
Fig. 8a (top),
8.
b (middle) and c (bottom).
Three
ways in which the pseudomonoclinic unit can be
stacked on (100) to produce different overall
diffraction effects.
TWINNED
TRICLINIC
MONOCLINIC
translating it by
b along a neighboring cell.
This is not exactly
true unless the cell is actually monoclinic, thus making a (triclinic)
equal to 90
.
The authors have taken x-ray photographs of several
twinned pectolite crystals from various localities and have found
that the diffraction effects substantiate the above ideas.
Fig. 9a
represents part of the composite reciprocal lattice of twinned
pectolite.
When the members of the twin are present in equal
amount, oscillating crystal photographs around b, as well as c-axis
precession photographs,
b.
show an apparent mirror plane normal to
Close examination of the films reveals, however, that the
registry of superposed spots is not perfect and that the registry is
slightly different in different specimens.
If the pseudomonoclinic cells are joined as in Fig. 8c, the
parawollastonite structure is formed having an oveidll monoclinic
symmetry with a doubling of the cell along a.
this monoclinic cell should be P2 /a
The symmetry of
with extra centers of
symmetry, not required by the space group , being present.
an attempt to refine the structure of parawollastonite,
In
however,
Tolliday (1958)2 found that the structure could not be refined using
Tolliday (1958) gives the systematic absences in the parawollastonite
diffraction pattern as: for hkl, 2h+k = 4n+2, and for OkO as k= 2n+1.
2 Tolliday, Joan (1958) .
Nature,
182, 1012-1013.
Crystal structure of p-wollastonite.
Figure 9
Figure 9a (left), 9b (middle) and 9c (right).
Reciprocal lattices
corresponding to the geometries shown in Figs. 8a, 8b, and 8c.
E
-A
(I
I
I
-9-
t
~I1
TWINNED
V
+ #+
O*
TRICLINIC
a*
MONOCLINIC
the centrosymmetric space group P2k/a and that when the noncentrosymmetric space P2
satisfactorily.
was assumed, the refinement proceeded
Since no coordinates,
structure factors or R-factors
were published, the validity of this assumption cannot be evaluated.
This is an extremely important point in the crystal chemistry of
CaSiO3 and one which should be resolved.
One of the interesting features of Figs. 9a, 9b, and 9c is that
the reci procal-lattice levels with k even are identical (when a = 90 )
and that the differences between triclinic, monoclinic and twinned
composite reciprocal lattices occur in only the odd levels.
It is
simpler for wollastonite than for pectolite to alternate to form the
parawollastonite type because a in wollastonite is 900 (within
experimental error) and because the symmetry of the repeating
unit in wollastonite is nearly monoclinic,
tetrahedra, as well as Ca
i.e., the Si
and Si2
and Ca 2 , are symmetrically equivalent.
The authors examined several wollastonite specimens from the
Monte Somma, Italy, Csiklova, Romania, and Crestmore, California,
localities before finding one which gave a parawollastonite diffraction
pattern.
Continuous radiation streaks along reciprocal lattice rows
parallel to a* were observed in the b-axis Weissenberg photographs
with k odd for the monoclinic as well as several triclinic specimens.
The parawollastonite crystal (locality:Crestmore,
California) was
supplied by Professor A. Pabst from a specimen, 138-80,
University of California collection.
in the
-I
:5
This effect was interpreted by Jeffery (1953)
as a type of disorder
due to the presence of regions of wollastonite and parawollastonite,
but it could also be thought of as an irregular sequence of pseudomonoclinic subcells.
Peacock (1935a)2 was unable to find any twinned triclinic
wollastonite in the material available to him.
Spencer (1903)3
however, reported a twinned wollastonite crystal from Chiapas,
Mexico, which would presumably give a diffraction pattern si.milar
I
to our twinned pectolite.
It is possible that some of the material
reported in the literature as ordered parawollastonite is actually
twinned wollastonite.
One could be misled by diffraction photographs
if the members of the twin were present in equal amounts.
One point which should be brought out here is that the way in
which these pseudomonoclinic units fit together results in what Ito
(1950)4 called "space-group twinning" and "structure twinning".
The "space-group twinning" is seen in the joining of pseudomonoclinic
cells to give an overall triclinic or a different monoclinic symmetry,
while "structure twinning" corresponds to our twinned pectolite.
This may be a valuable concept in an attempt to classify the triclinic
metasilicates.
1 Jeffery,
J.W. (1953).
wollastonite.
Unusual diffraction effects from a crystal of
Acta Cryst.,
6, 821-825.
2 Peacock,
M.A. (1935a) op. cit.
3 Spencer,
L.J. (1903).
4
Ito, T. (1950).
op. cit.
see paper by Collins, H.F. Min.Mag.,13, 356-362,
Conclusions
It has been established that wollastonite and pectolite have
many structural similarities, but yet must be classified as having
different structures.
Whether the pseudomonoclinic unit found in
both wollastonite and pectolite will be of significance in an overall
classification of the triclinic metasilicates will depend on the results
of structural investigation of such minerals as bustamite, rhodonite
and pyroxmangite.
This pseudomonoclinic unit at present appears to
be important in determining the relations between different modifications of a particular mineral in the wollastonite series.
Its departure
from actual monoclinic symmetry may also be a controlling factor in
determining what forms of a particular triclinic metasilicate can
occur.
Certainly, any study of the stability relations of wollastonite
and parawollastonite would have to be concerned with the details of
the structures 'to a greater extent than is usual in such investigations,
because the difference in energy between the two structures must be
quite small.
Acknowledgements
The authors wish to express thanks to Professor Clifford
Frondel and Dr. Alden Carpenter of Harvard University, Professor
Adolf Pabst of the University of California and Dr. George Switzer
of the U.S. National Museum for supplying much of the material
used in this investigation, and to Donald R. Peacor for many helpful
discussions.
Foundation.
This work was supported by the National Science
The computations were carried out at the M. I. T.
Computation Center.
Chapter III.
Drilling Coordinates for Crystal Structure Models
Abstract
Calculation of the spherical drilling coordinates,
p and <,
for crystal structure models often requires a prohibitive amount of
time, especially for structures of low symmetry.
The coordinates
are readily calculated using a vector algebraic method which is
applicable to any crystal system.
This method is designed for
computation with a digital computer.
INTRODUCTION
Many papers have described the construction of crystal
structure models
.
In those methods in which the structure is
represented by balls connected with rigid pins, it is necessary to
drill holes in the balls to maintain interatomic angles.
Several
drilling devices have been described2-4 which may be used to drill
these holes if the spherical drilling coordinates are known.
tion of these drilling coordinates,
p and <,
Calcula-
may be the most time-
consuming part of building a crystal structure model, especially
when the structure is one of low symmetry.
Because of this, the
structure is often "idealized" to save computation time, thus
obscuring important structural features.
A straightforward method of calculating drilling coordinates
for any crystal system has been developed.
This method is easily
programmed for a digital computer and has been used in the
construction of complex crystal structure models.
COORDINATE SYSTEMS
Computations are most easily carried out if atom positions
are known relative to an orthogonal coordinate system in which
coordinates are given directly in angstroms.
The transformation
60
of coordinates from a general axial system to an orthogonal
system
is given by
p=
xia + y. (b cos y) + z (c cos
q7 =
yi (b sin y) - z* (c M)
=
r
(1)
z. [ c(sin
2
p
-
M
2
where a, b, c, a, P and y are unit cell parameters of the original
axial system; x., y. and z
are fractional coordinates of atom i
relative to this system; p., q. and r.
L L
are coordinates of atom i based
-L
upon an orthogonal axial system, and
M = (cos
P cos y - cos a)/si
n y.
The coordinate axes are fixed so that, if L,
Z
and k are unit vectors
of the orthogonal coordinate system, i and a are colinear and i, ,
a and b are coplanar.
Consider a coordination group of n atoms, designated by
N. (i = 1,
2,.
..
,
n), which surrounds a central atom N
.
All atoms
are at the ends of vectors s. from the unit cell origin, where
2
2
2
s. = (p. + q. + r. )
L
The vector u. = s.
L
- s
the coordinating atoms.
L
L
.
(2)
is the vector from the central atom to one of
Figure 1 shows the spherical coordinate
system used in computing the drilling angles.
origin, the vector u
containing u
and u
Atom N
is at the
defines the direction p = 0, and the plane
defines the condition
coordinates of any atom N. , say N,
L
-3)
4
=
0.
The drilling
are computed from these
Figure 1
Coordination group of atoms N , N , N
-o -2' -3
atom N
,
and central
and its relation to the spherical coordinate system.
2:"-o
-
-
2:
Z3
-I
in-
Ze49
fixed positions.
CALCULATION OF p AND 4
For all atoms N. (i = 1,
-L -
atom N ,
-0'
U
2, ...
,
n) coordinating the central
.U
(3)
p. = cos
1
L
Figure 2 shows the vectors u
which is normal to u .
plane.
Let proj.
and u 3 projected onto a plane
Any angle 4., say 4,
L
3
is measured in this
-.
2 define the direction 4 = 0 in this plane.
Then
4. is the angle, taken counterclockwise, from proj. u to proj. u...
1
1".02
Consider a vector v. = u
to u
X u
.
All such vectors are normal
and therefore lie in the plane of Fig. 2.
angle between
and v.
y
~NL
Furthermore, the
is 4., whose magnitude may be determined
by
cos
where,
-i
v
v
2
2
.v
v
i
(4)
i
since 4 may vary from 0 to 27r, and since the cosine function
is unique only in the reglion 0 to 7r,
L
or
4,.
L
L
2r- 4.r.
L .
(5)
The ambiguity in the magnitude of 4. may be resolved in the
following way.
Consider the vector w = u x v 2
This vector lies
Figure 2
Vectors used in the calculation of angle <p3.
of the drawing is normal to vector
Plane
of Fig. 1.
.........................................................
......
..
...............
.......
.......
.............
......
..
....
....
....
.......
.................
....
..............
....................
..
.....
........
...........................
........................................
........................
.......................
......
......................
...................
..
.
..............
. ...........
............
................................
...............
.....................
n [id
...
.......
...........
.............
..........................................
...
.......
...................
...........
...................
.............
....................
...........
.....
.....
............
.......
... ...............
........
....................
................................................
...................
.......
........
........
..........
.........................
......................
..........................
....................
....................
.
..........
..............
...
Znf%
.............
.................
.........
.......
.
..... ..
.
..........
.................
...
.......
........
.......
..........
...............
...........
...................
.........................
..........
...
.........
...............
..................
...
.....
.............
.....................
X ............
..................
..........
...........................
.............
...........
........
.......
.....
. .
..........
.......................
..........
.....
..........
................
................
.....................
...............
.............
....
..........
.
.........
..
..........
..........
..................
66
in the plane of Fig. 2 and has a direction opposite to that of proj.
12.
~2'
The cosine of the angle 9 between v. and w is given by
.
V.
Cos
w
=
V.
(6)
w
L
If cos 0 > 0, V. lies in the shaded portion of Fig. 2 and
=
If cos 9 < 0, v. lies in the unshaded portion and *. = 2 r
L
L
DISCUSSION
This method was designed specifically for use with a digital
computer although it is certainly possible to use it with a desk
calculator.
In programming the computations it has been found
convenient to h4ve the interatomic distances u. printed out as a check
L
on the accuracy of input data.
In addition, results may be obtained
for enantiomorphically related coordination polyhedra.
coordinates of two such polyhedra,
_N.
and _N.,
The drilling
are related by
LL
p. = p.
PL 3
L
27
27r
*.=
(7)
-
4
In this laboratory, open structure models are usually built on
the scale of
half scale.
-
inch = 1A with the ball diameters at approximately
Using half-scale balls permits the interiors of the models
and the bonding directions to be seen more easily.
The balls are
67
joined by
inch brass rods cut so that the center-to-center distances
are proportional to the bond lengths.
ACKNOWLEDGEMENTS
This work was supported by a grant from the National Science
Foundation, and was done in part at the M. I. T. Computation Center,
Cambridge, Massachusetts.
The authors wish to thank Professor
M. J. Buerger for his helpful review of the manuscript.
68
REFERENCES
Deane K. Smith, Bibliography on molecular and crystal structure
models, National Bureau of Standards, Monograph 14,
Washington D.C.,
May 1960.
2 M. J. Buerger, Rev. Sci. Instr. 6, 412
(1935)
B. F. Decker and R. T. Asp, J. Chem. Educ. 32, 75 (1955)
C. A. Haywood, J. Sci. Instr. 26, 379 (1949)
r6
Chapter IV
Introduction
Although the gross details of the crystal structures of several
of the triclinic metasilicates have been known for several years, the
results of detailed refinements have just recently become available
(Chapter 1; Peacor, 1962).
These refinement results are extremely
useful for comparing related structures and are absolutely necessary
if one is interested in deducing the physical and chemical principles
upon which these structures are based.
Chapter I describes a
least-squares refinement of pectolite, Ca 2NaHSi 30 , which was
carried out with relatively crude intensity data originally used by
Buerger (1956) to solve the structure.
It was not possible to refine
R below 17% using this data (sets of hkO, Okl, and hOl reflections).
Because of the unusual interatomic distances which resulted from
this refinement, and because of the high R, it was felt that further
work should be done to improve the pectolite situation.
Further-
more, this would also provide another set of reliable data to be
used in comparing the different triclinic metasilicates.
Selection of material
A problem which arises when working with pectolite is that
it is quite hard to obtain a uniform single crystal.
The mineral
170
occurs in needles which tend to "feather" when broken.
end of a broken crystal consists of many smaller,
The feathered
somewhat mis-
oriented crystals which cause the major diffraction spots on a
Weissenberg film to be streaked parallel to the needle (when the
needle axis is the rotation axis).
When using a counter diffractometer,
this is especially troublesome because the long tails on the diffraction
peaks make the background corrections rather difficult.
A large number of pectolite samples from several localities
were searched for a crystal which would be small enough to be
entirely bathed by the x-ray beam and which would produce negligible
streaking in the diffraction photographs.
This was not achieved and,
instead, a long needle which would extend out of the x-ray beam at
all equi-inclination angles,
±e
, was chosen.
This crystal* was
transparent and bounded by distinct (100) and (001) faces with b as
the needle axis.
The crystal had the following dimensions:
between (100) and (100)
. 110 mm
between (001) and (001)
. 087 mm
needle length
3.68 mm
Even with this crystal, which was photographed with each end of the
needle out of the x-ray beam, the Weissenberg photographs showed
a slight streaking of the diffraction spots, indicating some misorientation of different regions of the crystal.
These were, however,
U.S.National Museum number 2452. Locality: Erie railroad cut,
Bergen Hill, New Jersey.
(Manchester, 1919).
'71
by far the best photographs obtained from any of the pectolite
crystals examined, although it appears that a penalty results from
this in that large primary extinction is evident in the observed
diffraction intensities (see the section on refinement).
An interesting result of the search for good single crystals
was that many of those examined were twins.
When the members
of the twin were present in equal amounts, b-axis oscillation
photographs showed an apparent mirror symmetry normal to b.
This is discussed in Chapter 11 in a comparison of twinned pectolite
and twinned wollastonite.
Apparently there are no published analyses of Bergen Hll
pectolite.
Schaller (1955),
however, gives several analyses for
pectolite including two for New Jersey pectolites, one from Paterson
and the other from Franklin.
Both of these analyses are given in
Table I along with the stochiometric composition.
It is likely that
the Bergen Hill material has a composition more like the Paterson
pectolite than the Franklin pectolite.
Unit cell data
Buerger (1956) published unit cell data on pectolite which was
obtained from precession films.
An attempt was made in the present
study to obtain precision Weissenberg photographs (Buerger, 1942)
from which measurements could be made and then refined with a
72
Table 1
Compositions of two pectolites and the hypothetical composition
computed from Ca 2NaHSi 30
Pater son, N. J.
(data from Schaller, 1955)
Franklin, N. J.
31.15
Hypothetical
CaO
33.20
MnO
.12
2. 57*
FeO
1.00
1.29
Na2 0
9.01
7.97
9.32
53.80
52.04
54.23
2.94
3.07
2.71
Si2
H20c
Etc.
33.74
2.12
100.07
*Includes 0.26 % ZnO
100.21
100,00
least-squares program written by Burnham (1962b).
pectolitets needlelike habit, however,
Because of
the only satisfactory photo-
graphs were ones taken with b as the rotation axis, thus giving refined
values of only a
,
c
,
and
p
Since these reciprocal cell parameters
compared very favorably with those of Buerger, the old values for
*
*
b , a ,and
y
*
*
were combined with the new for a,
a slightly different direct cell.
*
c ,and
p
*
to give
The old and new reciprocal and
direct cells are given in Table 2.
Space group
All previous investigators have assumed that the pectolite
space group is P1, although the author could find no experimental
evidence for a center of symmetry.
Peacock (1935b) does not
report any doubly terminated crystals for pectolite as he does for
wollastonite (Peacock, 1935a).
Although the refinement proceeded
satisfactorily assuming space group P1,
it would be interesting to
test pectolite for piezoelectricity and to try a cycle of refinement
assuming space group P1.
Intensity collection
Three-dimensi onal x-ray intensities were collected with a
Weissenberg counter diffractometer using CuK
radiation.
Ni foils
- -----
! -!,.,1,,,,__
__I-
-_
Table 2
Results of least-squares refinement of pectoli.te cell constants.
Buerger (1956)
.12878
A
Results of refinement
Adopted cell
0.128785+0.000002
.14549
.14310
0.143001+0.000003
88. 320
84. 58 0
84. 57808 +'0. 001060
77. 430
A
7. 99
7. 98818 +0.
7.04
7.03996
7.040
7.02
7.02468+0.00013
7.025
00011
7. 988
90. 520
90. 51984
90. 520
95. 180
95. 18062
95. 180
102.470
102.46865
102. 470
383.98
A3
* Refinement using precision Weissenberg data (59 observations) and Burnham (1962b) IBM 7090
program LCLSQ III.
COMM"M
at the x-ray tube and at the proportional counter aperture were used
in conjunction with a pulse height analyzer to discriminate against
unwanted radiation.
The proportional detector was a standard
Philips Xe-filled tube with preamplifier.
rate observed was about 9 X 103 c.p. s.
The largest counting
Background was counted
for 50 seconds on each side of a peak and the total intensity plus
background was obtained as the crystal rotated through a given
angle, A4>.
The integrated intensity, I, was obtained from
I = C - (B 1 + B 2 ) T/100
where C is the total count, B
and B2 are backgrounds, and T is
the time required to collect C.
Structure factors were obtained using data reduction programs
written by Burnham (1962a) and Prewitt (1960).
This included a
prismatic crystal absorption correction in which path lengths for
the zero level were computed for all reflections and then divided
by cos p
.
replaced by
In addition, the linear absorption coefficient,
I/cos '
1
,was
to compensate for the increased scattering
volume for upper levels due to the use of an "infinite" crystal.
A
total of 1357 structure factors were thus produced.
Refi nement
All refinement was carried out using the IBM 7090 program,
SFLSQ3, described in Appendix I.
Starting coordinates were those
...........
given in Table I of Chapter 1.
Form factors for atoms assumed to be
half-ionized were taken from Freeman (1959),
and correction was made
for both the real and imaginary anomalous scattering factors for Ca
(International Tables (1962),
Vol.
II).
The first cycles of refinement
used only reflections from the reciprocal lattice levels 0, 1, 2, and 3 in
order to save computing time.
After an initial cycle to adjust the scale
factor, the initial R-factor was 0. 174, which is, incidentally, about
the same as was found using the original Buerger data and the same
atom coordinates.
In a further cycle in which all atom coordinates
plus the overall scale factor were varied, the R went up to 0. 239.
This result made the statistical weighting scheme (Burnham, 1962a)
suspect, especially since fT (F
reflections.
- F ) varied quite widely for different
In the next two cycles, the original coordinates were
used as a starting point and a weighting scheme recommended by
Cruickshank (1961) was incorporated into the program where
(Vw=
1/(a+F +C F o 2
0
In this expression, a = 2F.
me n
and C = 2/F
max
.
At the end of the
second cycle the R was 0.085, thus supporting the assumption that
the weighting in the original cycles was incorrect.
In order to check
whether the Burnham weighting scheme itself was causing trouble or
whether the philosophy of weights based only on counting statistics
alone was in error, two more cycles were tried using the Evans (1961)
weighting scheme which is somewhat different from that of Burnham.
At the end of the two cycles, the R had gone up to 0. 148.
After this,
all weighting was carried out using the Cruickshank method.
No
analysis has been made of why trouble was encountered with the
Burnham and Evans methods, but it appears that there must be
systematic errors present in the data which are not accounted for
in schemes based solely on counting statistics.
In the next cycle, temperature factors in addition to atom
coordinates were varied, resulting in an R of 0.079.
however,
0
negative temperature factors for atoms SL
There were,
Si.2'
3,
and
Another cycle was run in which reflections were rejected from
the refinement but included in the R-factor.
The R was reduced to
0.075 and all temperature factors refined to positive values.
A
further cycle using all 1357 reflections but with the rejection test,
left R at 0.075.
It is noteworthy that for the strongest reflections, the
calculated value is much larger than the observed, indicating the
need for a primary extinction correction.
Although none was made,
it should be done before this work is published.
A possible
correction to the observed intensities is
I C= 1I /(1 - t 1I )
'c
o/(-tI
where t is a very small number determined by comparing the
present I and I .
-o
-c
The need for this correction is supported by
the fact that removing the two strongest reflections (120 and 220)
reduces R to 0. 070 and that removing the twenty strongest reduces
R to about 0. 05.
It is probable that inclusion of a hydrogen contribu-
tion in the calculated structure factor plus anisotropic temperature
factor refinement will lower the R even further.
The final atom coordinates and isotropic temperature factors
are given in Table 3.
It is doubtful that there will be any significant
change in these coordinates in further refinement.
Evaluation of the structure
Figure 1 is a projection of the pectolite structure onto (101)
of the pseudomonoclinic pectolite cell described in Chapter It.
The
structure is made up of double columns of edge-sharing Ca octahedra which extend in the b-direction.
Adjoining sets of octahedra
are linked by single silicate chains and irregularly coordinated
Na atoms.
The silicate chains share only corners with the Ca
octahedra, but five edges of the Na polyhedron are shared with the
silicate tetrahedra.
This is in contrast with the wollastonite
structure where the Ca octahedra are found in columns of three
edge-sharing octahedra wide with the Ca
and Ca
octahedra each
sharing one edge with the Si.3 tetrahedron, and the Ca3 octahedron
sharing edges with only the other octahedra.
Table 4 is a compilation of the interatomic distances in
pectolite, computed using the coordinates of Table 3.
Each silicon
70
Table 3
Refined coordinates for pectolite
Atom
x
y
z
B
Ca 1
.8549
.5939
.1449
.39
Ca2
.8465
.0837
.1404
.38
Na
.5520
.2591
.3430
.83
Si
.2182
.4010
.3373
.20
Si2
.2152
.9553
.3446
.23
Si3
.4505
.7353
.1448
.06
0
.6520
.7863
.1296
.45
02
.3299
.7036
-. 0524
.24
03
.1861
.5010
.5385
.30
04
.1782
.8433
.5406
.40
05
.0626
.3853
.1742
.34
06
.0598
.8958
.1775
.44
07
.3987
.5411
.2720
.38
08
.3952
.9044
.2750
.37
09
.2630
.1934
.3861
.32
8;
Figure 1
Pectolite structure projected onto (101).
The Ca
locations and coordination are represented by
octahedra, but, because of its irregular coordination, Na is represented by both a shaded circle
and, for a single sodium atom, an irregular
polyhedron.
Table 4
Interatomic distances in pectolite
SL
Si
Si
2
3
03
1.638
05
1. 597
07
1.666
09
1.611
Av.
1. 628
04
1.612
06
1. 612
0 8
1.668
09
1.654
Av.
1.637
0
1.585
02
07
0 8
Ca 4
1.599
1.638
1.652
Av.
1.619
01
2.322
02
2.336
03
2.359
05
2.439
05
2. 390
A
Ca
Na
O
2
06
2. 381
Av.
2.371
0
2.318
02
2.320
0
2.330
05
2.425
06
2.414
06:
2.3
Av.
Z-(o
0 2
2.319
03
2.461
04
2.489
07
2.578
0 7
2.991
0
2.560
08 T
2.961
0
2.303
Av.
2.583 (2.452 excluding 0 7 and 08
02
2.709
07
2.639
08
2.654
02
07
2.626
2.639
03
3r
2. 971
04
2.424
05
2.710
07
2. 616
09
2.620
2.696
04
2.636
09
05
07
09
2.678
2.695
2.673
Os
07
08
2.692
09
2. 687
08
2.564
09
2.618
0
2.629
is coordinated by four oxygen atoms, the average Si-O distance being
1. 628
A,
a little higher than the "ideal" metasilicate distance of
about 1.622 given by Smith and Bailey (1962).
The range from the
smallest (1.585A) to the largest (1.668A) Si-O distance is 0.083A,
as large a spread as Burnham (1962) found in sillimanite, where
there is supposed to be "drastic" shortening of shared polyhedron
edges.
Ca
and Ca 2 are octahedrally coordinated by 6 oxygens at
average distances of 2. 371A and 2. 362
A,
respecti vely.
Na is
irregularly surrounded by 6 oxygens at an average distance of
2.452
A
with two more at 2. 991
A and
2. 961
A.
Table 5 gives some of the interatomic distances rearranged
to show how each oxygen is coordinated by cations.
Because of the
complicated way in which the Na polyhedron shares edges with the
silicate tetrahedra, it is not as easy to divide the oxygen atoms
into different "types" according to their coordination.
In addition,
until the hydrogen location is proven, a realistic division cannot be
made.
In general, however, 0
through 06 can be characterized
by being coordinated by one Si and two or three cations while 07
through 09 are coordinated by two Si and one or two cations.
When
an oxygen is coordinated by two Si, the Si-O distances are considerably longer than when the oxygen is coordinated by only one Si.
As was stated in Chapter II, the most likely location for
H is between 03 and 0
distance of 2.424
A.
because of the anomalously short 0 3-04
This, however,
is the edge of the Si3
Table 5
Oxygen coordination in pectolite
0
02
Si3
1.585
Ca 1
2.322
Ca
2.318
SI
2
1
Ca
2
Na
03
Ca 2
1
1
1.666
2. 320
Si.3
1.638
2. 319
Na
2.578
Nar
2.991
Si
1.668
Si.3
1.652
Na
2.560
Na'
2.961
Si
1.611
07
2.359
08
1.612
2
Ca
2
2.330
2.489
Na
1.597
SLi
Ca
09
Si
2
Na
H?
0 5
2.414
Si
2. 322
HI
Si
2. 381
Ca2
2.461
Na
04
1.612
1.638
Si
Ca
Si 2
Ca 1
1. -') -)
3
Ca
06
1
2.439
Ca 1
2.390
Ca
2.425
2
1.654
2.303
r'7
tetrahedron shared by the Na polyhedron.
Difference maps will have
to be computed to show whether hydrogen is actually present between
03 and 04.
It is interesting to note that all shared polyhedron edges are
shorter than unshared ones,
in accordance with Pauling t s rules.
The unshared tetrahedral edges are in the range of 2. 673
2.710
1
1
while the shared edges range from 2. 564 I to 2. 636
to
1.
Conclusions
There is no doubt that we now have a reasonably accurate
set of interatomic distances for pectolite, even though a few additional
improvements could be made by including a correction for primary
extinction and by refining anisotropic temperature factors.
In
addition, some investigation should be made of the possibility that
the Na position contains more scattering power than has been
assumed, i. e., a small percentage of heavier atoms.
This is
supported by the large temperature factor of Na (B = 0. 83).
-9
References
Buerger, Martin J. (1942),
X-ray Crystallography. John Wiley and
Sons, New York.
Buerger, M.J. (1956),
The determination of the crystal structure
of pectolite, Ca 2NaHSi 30
Burnham, Charles W. (1962a),
.
Z. Krist.,
108, 248-261.
The structures and crystal chemistry
of the aluminum-silicate minerals.
Ph.D. Thesis, M. I. T. ,
Cambridge, Mass.
Burnham, Charles W. (1962b),
Crystallographic lattice constant
least-squtares refinement program, LCLSQ (Mark III).
Unpublished IBM 7090 computer program.
Cruickshank, D.W.J.,
and Mary R.
Deana E. Pilling, A.
Truter (1961),
Bujosa, F.M. Lovell,
In Computing methods and the phase
problem in x-ray crystal analysis.
Pergamon Press, Oxford,
32-78.
Evans, Howard T.,
Jr. (1961),
Weighting factors for single-crystal
x-ray diffraction intensity data.
Acta Cryst. 14, 689, :
.
Freeman, A.S. (1959), Atomic scattering factors for spherical and
aspherical charge distributions.
Acta Cryst. 12, 261-271.
International Tables for X-ray Crystallography,
Vol. III (1962).
The Kynoch Press, Birmingham, England.
Manchester, James G. (1919),
Am. Mineral.,
The minerals of the Bergen archways.
4, 107-116.
Peacock, M.A. (1935a), On wollastonite and parawollastonite.
Am.J.Sci.,
30, 495-529.
Peacock, M.A. (1935b), On pectolite. Z. Krist.,
90,
97-111.
Peacor, Donald R. (1962),
The structures and crystal chemistry of
bustamite and rhodonite. Ph.D. Thesis, M. I.T.,
Prewitt, Charles T. (1960),
Cambridge, Mass.
The parameters T and 4> for equi-inclination,
with application to the single-crystal counter diffractometer.
Z. Krist.,
114, 356-360.
Schaller, Waldemar T. (1955),
series. Am. Mineral.,
40,
The pectolite-schizolite-serandite
1022-1031.
-
0C
- . 7
4
9
-
-
4s e02
n1
Chapter V
Review of the crystallographic investigation of metasiLicates
The metasilicates are classically grouped together because
of their characteristic chemical composition which is hypothetically
derived from metasilic acid, H 2SiO
.
This is not entirely satisfac-
tory because the structures of the minerals classed as metasilicates
can be quite different from one another. A classification based on
structure was proposed by Strunz (1941) which uses Greek prefixes
to describe the way in which the silicate tetrahedra are linked
together.
This classification divides metasilicates into two cate-
gorLes, cyclosilicates (rings of tetrahedra) and inosilicates (chains
of tetrahedra).
Inosilicates, however,
which are not metasilicates, e.g.,
contain additional members
amphiboles.
The present chapter,
therefore, discusses ino-metasilicates, which can be further defined
as compounds of formula Am (SiO 3n) containing single chains of
silicate tetrahedra.
Table I is a compilation of the more common members of
the ino-metasilicate group and shows the current status of structural
work on each entry.
Although only a few of these structures have
been refined in detail, the gross relations between the minerals are
much clearer today than they were just a few years ago.
Table I
List of crystallographically interesting metasilicates
Name
enstatite
Formula
MgSiO
MgSi.O
P2 1 /c
3
protoenstatite
MgSi0
Pbcn?
3
p igeonite
(Ca, Mg, Fe)SiO3
P2 /c
augite
(Ca, Mg, Fe)Si.O
3
C2/c
diops ide
CaMgSi.
hedenbergite
CaFeSi
acmite
NaFeSi~ O
jade ite
NaAlSi2 06
it
spodumene
LiAISi2 06
ft
wollastonite
CaSiO
parawollastonite
CaSiO
bustamite
CaMnSi2 0
Al
rhodonite
pyroxmang ite
CaMn5 6 018
(Mn, Fe, Ca, Mg)Si.O
3
pectolite
Ca 2NaHSi30
P!
Pi
P!
serandite
Mn 2NaHSi3 0
P!.
babingtonite
(Ca, Fe+2, Fe 3)S3
P!
2
B - inferred or incomplete
C - structure correct
D - structure refined
x
tI
johannsenite
A - not known
x
06
06
CaMnSi2 06
Key to status of structure:
Status of
structure
A B C D
Pbca
3
clinoenstatite
Space group
It
It
6
x x
3
3
P2 1 /a
x
x
Status of structural work.
The first pyroxene structure to be
solved by x-ray methods was that of diopside (Warren and Bragg, 1928).
This was followed closely by two more papers on the structures of
pyroxenes,
the first on enstatite (Warren and Modell, 1930) and the
other on the monoclinic pyroxenes (Warren and Biscoe, 1931).
In
the latter paper, Warren and Biscoe compared diffraction patterns
of diopside with those of hedenbergite, augite, clinoenstatite, acmite,
jadeite, and spodumene and concluded that they were all very
similar in crystal structure.
An attempt was also made to solve
the structures of wollastonite and pectolite, but since their structures
were quite unlike that of diopside, this attempt did not succeed.
A few years later Barnick (1935) proposed a ring structure
for parawollastonite which has since been proven to be wrong.
As
will be pointed out in Chapter VI, this choice of material was
unfortunate.
He should have worked with the triclinic structure,
wollastonite,
instead, because parawollastonite is a twinned
wollastonite.
Two important papers (Peacock, 1935a;1935b) on wollastonite
t
and pectolite appeared at about the same time as Barnick s paper.
Although these papers did not directly shed now light on the structures
of these minerals, they have been proven to be very valuable as a
source of morphological and optical data.
Warren and Biscoe (1931)
had shown that both the wollastonite and the pectolite they had
there
examined were triclinic, and Peacock showed further that
were actually two forms of wollastonite which were being confused.
He called the triclinic phase wollastonite and the monoclinic one
parawollastonite.
He further named the high temperature form
first described by Bourgeois (1882) and which was not being
confused with wollastonite, pseudowollastonite.
No further structural work was done until Ito (1950) tried
to explain the differences between wollastonite and parawollastoni.te.
Since he based much of his argument on the assumption that the
Barnick structure was correct, this work has not been given proper
credit for the portion that really described the basis for distinguishing wollastonite and parawollastonite (Chapters II and VI).
In 1956,
structures for pectolite and wollastonite were
published which have subsequently been found to be correct
(Chapters I and II).
Shortly thereafter,
reinvestigations of the
Mg-Fe-Ca pyroxenes showed that the space groups of clinoenstatite
and pigeonite were different from that of diopside(P2,/c instead of
2/c) and that protoenstatite was a distinct phase with an apparently
different structure from the other enstatites.
Morimoto, Appleman,
and Evans (1960) refined the structures of clinoenstatite and
pigeorite, and Lindemann (1961) refined enstatite, thus giving
fairly reliable sets of coordinates for these pyroxenes.
refinements of diop side, acmite,
No
spodumene, jadeite, hedenbergite,
but it is probable
or j-ohannsenite are known at the present time,
that these will be refined in the near future.
No minerals are known with the pectolite structure except
schizolite and serandite, which involve substitution of Mn for Ca in
pectolite.
The ideal formula for schizolite is CaMnNaHSi 30
and
for serandite is Mn NaHSi 309, but as far as has been determined,
a continuous series exists from pectolite to a phase containing about
77% of the manganese end member (Schaller, 1955).
There may be
more to this problem than one might suppose, since Warren and
Biscoe (1931) found that the c axis in schizolite (locality not given)
was double that of c in pectolite.
This is exactly the same relation
as bustamite has to wollastonite and consequently bears further
investigation.
However, Ito (1939) and Liebau (1957) give cell
constants which are approximately those of pectolite.
Peacor (1962) has refined the structures of bustamite and
rhodonite, thus providing detailed information about these structures.
This leaves the pyroxmangite type structure as the only category of
Table 1 in which no detailed refinement has been done although
Liebau (1959) proposed a structure but did not give any coordinates.
One
Future possibilities for ino-metas ilicate investigations.
of the most probable things that will be found when metasilicates
are further investigated is that additional phases will be found which
are a result of space group twinning (Chapter VI).
The hydrated
calcium silicates will also be of considerable interest if any are
shown to definitely have single silicate chains.
communication) thinks that babingtonite,
Peacor (personal
(Ca, Fe +2,
Fe3 )SO
3
may
have the rhodonite structure and it would be very interesting to see
whether other structures like those of rhodonite and pyroxmangite
could be found.
Another topic of interest for the future would be the relating
of the physical and chemical properties of the minerals to their
structures.
This has already been done to some extent, but a
number of things are left to be explained in terms of the structures.
These include cleavage, optical properties, chemical bonding,
mechanisms of phase changes, twinning, and crystal morphology.
The ultimate as far as the mineralogist or petrologist is concerned
is to be able to relate natural phase assemblages with conditions
necessary for their formation.
This, of course, involves much more
than the ino-metasilicates and probably will require many years of
work before a satisfactory solution is found.
References
Barnick, Max A. W. (1935),
Wollastonite.
Strukturuntesuchung des nattirlichen
Mitt. Kaiser-Wilhelm-Inst. Silikat forsch.,
No. 172,
1-36.
Bourgeois, L.
(1882), Essai de production artificielle de wollastonite
et de meionite.
Brown, W.L.,
Bull. Soc. Min. France, 5,
13-16.
N. Morimoto, and J.V.Smith (1961),
A structural
explanation of the polymorphism and transitions
of MgSiO3'
J. Geol., 69, 609-616.
Ito, T. (1939),
The determination of lattice constants of triclinic
crystals from one crystals setting- a special case.
Z. Krist.,
100, 437-439.
Liebau, Friedrich (1957), Uber die Struktur des Schizoliths.
Mh.,
Neves Jb. Mineral.,
Lindemann, W. (1961),
Jahr. Min.,
10, 227-229.
Bertrag zur Enstatit struktur.
Neves
10, 226-233.
Peacock, M.A. (1935a), On wollastonite and parawollastonite.
Am. J. Sci.,
30, 495-529.
Peacock, M.A. (1935b),
On pectolite-.
Z. Krist.,
90, 7-111.
Peacor, Donald R. (1962), Structures and crystal chemistry of
bustamite and rhodonite.
Schaller, Waldemar T. (1955),
series.
Am. Min.,
Smith, J.V. (1959),
Acta Cryst.,
M.I. T. Ph.D. Thesis.
The pectolite-schizolite-serandite
40, 1022-1031.
The crystal structure of proto-enstatite, MgSiO3'
12, 515-519.
Strunz, Hugo (1941),
Mineral. Tabellen. Akademische Verlags-
gesellschaft Leipzig.
S4
Warren, B.E. and J. BLscoe (1931),
monoclini.c pyroxenes.
Warren, B.,
Z. Krist.,
80, 391-401.
and W. Lawrence Bragg (1928),
diopsLde, CaMg(SLO3 )2.
Warren, B. E.,
MgSLO 3 .
The crystal structure of the
Z. Krist.,
and D. I. Modell (1930),
Z. Krist.,
75, 1-14.
The structure of
69, 168-193.
The structure of enstatite
Chapter VI
Twinning in silicate crystallography
The presence of twinning is probably the most common cause
of experimental difficulty in the field of silicate crystallography.
It
is obvious that one can expect to have trouble solving or refining a
crystal structure if the material being investigated is twinned,
although techniques have been developed (Morimoto, Appleman, and
Evans, 1960; Frueh, 1962) which will give reasonable results if the
nature of the twinning is known.
It is sometimes difficult, however,
to establish what twin laws are operating in a crystal, or that
twinning exists at all.
It will be the purpose of this chapter to look
into some of the aspects of twinning in silicate minerals, particularly
in the light of experience gained in working with wollastonite and
parawollastonite as described in Chapter II.
Structure and space group twinning.
It is not generally
realized that in addition to the macroscopic twinning of crystals
there is another type which takes place on unit cell scale.
This
latter type produces a reduction or enhancement of the symmetry
of the crystal over the symmetry of some basic repeating unit.
Ito (1950) called these structure twinning and space group twinning.
Structure twinning is represented by all examples of twinning
given in, say, a standard mineralogy textbook.
Space group
twinning, however, manifests itself in a much more subtle way and
can usually only be detected as such by interpretation of x-ray diffraction patterns.
Good and perhaps classic examples of space group
twinning are found in wollastonite and parawollastonite as discussed
in Chapter II.
Other possibilities for space group twinning will be
discussed below.
One problem which arises when space group twinning is considered is how one distinguishes very fine polysynthetic twinning from
actual space group twinning, since the two are quite similar except
for the scale of the repetition.
of the diffracted x-rays; that is,
The answer seems to be in the nature
since our only tool for comparing
these phenomena is x-ray diffraction, then the criteria for distinguishing between them are to be found in differences in diffraction effects.
If, for example, the observed diffraction pattern is an apparent
result of space group twinning, the resulting structure must be accepted as having the symmetry of the observed space group even
though it is possible that polysynthetic twinning is present on a very
fine scale.
If, however, a method is devised which will distinguish
between these two alternatives, then one or the other can be adopted.
The exact nature of the space group twin does not concern us here.
What is important is that the repetition is on a fine enough scale so
that x-ray coherence is such that the diffraction pattern has an apof the
parent symmetry and that this symmetry is representative
crystal structure in question.
101
Implications of space group twinning. Aside from academic
interest in this phenomena, the most important result of understanding the nature of space group twinning lies in the phase
equilibria of a system containing a compound which exhibits space
group twinning.
Take CaSi.O3 , for example.
Triclinic wollastonite
has been demonstrated to have a definite stability field, but parawollastonite has not been synthesized and probably forms only under
very unique conditions.
From this standpoint, parawollastonite
appears to be a metastable phase under most conditions and may
not even have a stability field at all.
Figure 1 demonstrates the geometrical relationships between
wollastonite and parawollastonite.
In the upper left of the diagram,
the wollastonite pseudomonoclinic subcell described in Chapter II is
represented by a cell containing two triangles on mirror planes
at y =
13
-and
y =
-.
Each of these triangles represents a silicate
chain in the actual structure.
There are no symmetry elements
other than the single center and the two mirror planes.
If the
other identical cells are added in the y direction, then centers
1
of symmetry and a 2 1 axis at x = -2 relate the cells to one
another.
Now, Fig. 2 shows that it is possible to join
as
wollastonite cells on (100) in one of two ways which, as far
first coordination spheres are concerned, are geometrically
identical (provided the symmetry of the subcell is truly monoclinic).
1
This stacking Involves a shift of +-.b,
resulting in an overall triclinic
102-
Figure i
Schematic representations of the ways in which
pseudomonoclinic subcells can be joined to give a
different overall symmetry.
The diagram in the
upper left shows how the subcells can be stacked
on (010) to give space group P2,/n.
On the right,
the cells are joined on (100) in successive shifts
of +lb or -1b to give obverse and reverse P!
symmetry (wollastonite).
In the lower left, cells
are joined with alternating shifts of + and -- b to
give overall P2 1 /a
symmetry (parawollastonite).
TRICLINIC
+
0
~~~~~~-0
a
TRICLINIC
MONOCLINIC
i
a sin /
I
Figure 2
Pseudomonoclinic CaSi.O3 subcells joined on (100) to
give wollastonite (top) and parawollastonite (bottom).
TRICLINIC
+
0
0
+
0
TRICLINIC
MONOCLINIC
|
a sin/3
I
1±6
symmetry.
Returning to Fig. 1, the stacking of subcells with
successive shifts of +ib
of -- b,
gives the same triclinic cell as do shifts
except that the cells are oriented differently in space.
Parawollastonite is formed by adding cells on (100) with
alternating shifts of +-b and -- b.
4at x =
When this is done, the centers
in the single cell still exist for the single cell, but do not
relate adjoining cells with each other.
Instead, the centers which
arise between subcells define a new cell with symmetry p2
1
/a.
In order to check the validity of the above findings, structure
factors were calculated for each structure with a single atom placed
on the mirror plane of the subcell.
The resulting patterns exactly
duplicated the extinction rules observed in wollastonite and parawollastonite, respectively.
It is interesting to compare these results with the bustamite
structure which has been solved by Peacor (1962).
Figure 3 shows
the two ways in which the wollastonite pseudomonoclinic subcells can
be joined on (001) with shifts of either +-b or -- b.
22Here, because of the cell geometry, the mirror planes are
continuous across the cells.
The sequence in the upper part of the
figure produces the wollastonite structure and the one in the lower
part produces the bustamite structure.
This is really the only
difference between bustamite and wollastonite.
however,
This does result,
in different coordinations for two-thirds of the large
cations in each of the structures.
Figure 4 shows how the actual
Figure 3
Schematic representations of the wollastonite (top) and
bustamite (bottom) structure.
The relation between
wollastonite and bustamite is similar to that between
wollastonite and parawollastonite except that successive
pseudomonoclinic subcells are not shifted on (001) in
wollastonite and are shifted by + or -- b in bustamite.
WOLLASTONIT E
c sinl,
BUSTAMITE
c sin/3
Figure 4
Result of joining CaSiO3 subcells as described in
Figure 3 to give the wollastonite structure (top) and
the bustamite structure (bottom).
WOLLASTONI TE
BUSTAMITE
lec
sine-
structures look when joined in this way.
It is interesting to note that
although bustamite has an A-centered cell, the extinction rule is
compatible with space group extinction rules, but this is not so with
parawollastonite.
This occurs because the bustamite structure does
not contain the "leftover" centers of symmetry that are found in
parawollastonite.
Also the bustamite structure is much more
different from wollastonite than is parawollastonite.
Energy relations in parawollastonite and bustamite.
There
must be some way of rationalizing the foregoing observations in terms
of energy.
To provide a means of describing the ways in which cells
can be joined, the letter A can be assigned to an interface with a
shift of, say, +. b, and B to one with a shift of -- b.
4-
4-
wollastonite would be represented by AAAAA...
parawollastonite by ABABAB...
.
Then
or BBBBB. ..
and
Now, ignoring the mechanism of
twin formation, we can say that in the ideal case, there is no
difference in energy between the interfaces A and B.
must be between AA and AB.
The difference
It is probable that under most con-
ditions,thea6ionition;AA, nrBB has a lower energy than AB, and
therefore wollastonite is the stable phase.
Other conditions,
such
as mechanical stress, could favor AB and parawollastonite would
be formed.
On the other hand, bustamite seems to favor the AB
sequence to the exclusion of AA.
It would be interesting to try
calculating these different energies using just Coulombic interaction
such as was done by Zoltai and Buerger (1960) on rings of tetrahedra.
Space group twinning in other silicates. There are other silicates which, although common, have not been successfully shown to
have definite stability fields in the laboratory.
Boyd (1959) states
that conditions for stability of anthophyllite have not been found.
Ito (1950) tried to show that the orthorhombic anthophyllite structure
was the result of space group twinning of the tremolite (or cummingtonite) structure, but was unsuccessful.
There are, however,
cer-
tain relations between these structures which bear further investigation and which may shed light on the problems of the Mg-Fe
amphi bole s.
There is more evidence for a space group twinning relationship between the ortho and clino pyroxenes.
Strunz (1958) states
that the orthorhombic pyroxene structures arise from the monoclinic
through a space lattice twinning on a(100).
Brown, Morimoto, and
Smith (1961) show that although the orientations with respect to the
silicate chains are different, the structures of protoenstatite,
clinoenstatite and rhombic enstatite differ in the same way as do
wollastonite and parawollastonite.
In the notation given above for
the interfaces of adjoining cells, these pyroxenes are represented as
clinoenstatite
AAAAAA
(wollastonite)
protoenstatite
ABABAB
(parawollastonite)
enstatite
AABBAA
(no known equivalent)
It is interesting that the phase relations between these minerals are
as yet unclear (Brown, Morimoto, and Smith, 1961) because of the
tendency for these minerals to exist metastably.
Energy calculations
could well be introduced here.
There are many parallels in the feldspar problem to what is
found in wollastonite-parawollastonite,
feldspars are much more complex.
although apparently the
Feldspar diffraction patterns
contain extra spots, diffuse spots, and streaks which are also observed in the wollastonites.
Although the feldspar situation is much
too involved to be discussed here,
it should be noted that there seems
to be a common thread running through all of the structual discussions
of feldspars which is not yet clearly defined.
This, too, is a fruitful
field for research.
One thing that is common to all silicate structures is that
there is a tendency for the oxygen to form close-packed, although
distorted, arrangements.
This is more evident in olivine than in
quartz, but it is to some extent true in all silicate structures.
The
analogy here is to stacking faults seen in close-packed structures
such as those of the metals and the fact that a close-packed plane
offers different sites for the location of adjoining atoms which differ
little in energy.
It may well be that the location of a twin boundary
in the silicates will often be one of these close-packed planes.
This
point is not clear in parawollastonite because (100) is not strictly
a plane of close packing, but the ease of demonstrating the twinning
on (100) may be misleading and the real differences may occur
elsewhere.
It is true, however, that there is a definite boundary
in parawollastonite which is common to adjoining cells and that this
boundary plane contains only oxygen atoms.
It is quite evident that while much work has been done on
twinning of the silicates, not much of it has concerned the actual
structures.
There is much room for descriptive investigations to
compare the similarities and differences among different twinned
silicates and for analytical investigations which can justify why a
particular twin occurs.
I
References
Boyd, Francis R. (1959),
Hydrothermal investigations of amphiboles.
Researches in geochemistry, P.H.Abelson, Ed. John Wiley and
Sons, New York.
Brown, W. L.,
N. Morimoto, and J.V. Smith (1961), A structaral
explanation of the polymorphism and transitions of
MgSiO3'
J. Geol., 69, 609-616.
Ito, T. (1950),
X-Ray Studies in Polymorphism.
Maruzen, Tokyo.
Morimoto, Nobuo, Daniel E. Appleman, and Howard T. Evans, Jr.
(1960),
The crystal structures of clinoenstatite and pigeonite.
Z. Krist.,
114, 120-147.
Strunz, Hugo (1957),
Mineralogische tabellen,
Akademische
Verlagsgesellschaft, Leipzig.
Zoltai, Tibor, and M. J.Buerger (1960),
rings of tetrahedra.
L
Z. Kr st.,
The relative energies of
114, 1-8.
Chapter VII
Crystal structures related to wollastonite and pectolite
Liebau (1957;1961) has made extensive compilations of
inorganic silicates, germanates, phosphates, arsenates, vanadates,
and fluoberyllates whose structures contain chains of tetrahedra.
Although there appears to be a great number of these compounds, all
fall into relatively few classes, one of which is represented by the
wollastonite structure.
No attempt will be made here to reproduce
or improve this extensive tabulated data except to give examples of
each class and to comment briefly on the status of structural information for each of these classes.
Table 1 is a listing of the different structure types as
proposed by Liebau.
Each structure type in this table is character-
ized by the number of tetrahedra in the repeating unit of the chain.
For example, the pyroxene and the wollastonite structures have
repeats of two and three tetrahedra, respectively, whereas
rhodonite and pyroxmangite contain five and seven tetrahedra,
respectively.
The pseudowollastonite structure contains chains of
three tetrahedra li.nked together to form a three-membered ring.
Status of structural information.
One of the most
interesting aspects of the classification is that here is a series of
compounds which vary in structure primarily because of differences
in atom radii.
With enough information about the structures, one
Table I
Classification of compounds with formula ABX 3 (after Liebau, 1961)
Number of
tetrahedra
in repeat
unit
Compound
CuGeO
3
Cation
radius
ratio
. 99
Remarks
CuSiO3 not known
MgSiO 3 (enstatite)
1.60
well refined
CaMg(SiO 3 ) 2 (diopside)
1.98
structure known
but not refined
7
(Ca,Mg) (Mn,Fe) (SiO
3) 7
6
(pyroxmangite)
1.98
structure proposed
but not refined
5
CaMn 5 (SiO 3 ) 6 (rhodonite)
1.98
well refi ned
3
CaSiO 3 (wollastonite)
2.36
well refined
CaMn(SiO 3 ) 2 (bustamite)
2.14
well refined
CaSi.O3 (pseudowollastonite)
2.36
SrSiO 3
2.67
structure proposed
but not proven
or refined
SrGeO
1.53
2
co (3-
member ed
r ing)
3
2
BaSiO
3.19
3
BaGeO
1 .84
3
structure proposed
but not proven
might be able to relate crystal chemistry with phase equilibria in a
quantitative way.
It might be possible, for instance, to predict the
temperature at which wolastonite transforms to pseudowollastonite
by knowing how the effective radii of Ca, Si, and 0 vary with
temperature.
The cation radius ratios given in Table i show that
there is a considerable range in the radius ratio among the different
structural types, although this is apparently not the only factor
which controls the structure of a particular phase.
Undoubtedly,
quantitative work would require that the radii of all atoms be known
accurately at different temperatures.
At present, only a few structures in the groups of -,Table 1 are
known with any degree of accuracy, some are very poorly known, and
no work at all has been done at other than room temperature.
The
status of the pyroxenes was discussed in Chapter V, there being
refined coordinates for only enstatite, clinoenstatite, and pigeonite.
In the other groups, only wollastonite, bustamite, and rhodonite
have been adequately refined.
Structures have been proposed for CuGeO 3 (Ginetti, 1954),
pseudowollastonite (Dornberger-Schiff, 1962), BaSiO3 (Lazarev et al.,
1962), and pyroxmangite (Liebau, 1959), but none have been refined
and some may even be wrong.
Certainly, there is much room for
further work in this series of compounds.
Pseudowollastonite.
Since pseudowollastonite is very closely
related to the subject of this thesis, some important features of the
status of its structure should be noted here.
It was originally
intended that some experimental work would be done on this structure
but, due to the difficulty of obtaining good single crystals for
Since Hilmer (1958) had
structure refinement, this was not done.
published a note on the structure of SrGeO3 which is thought to have
about the same structure as pseudowollastonite,
the prospect of
publishing a proposed pseudowollastonite structure without a refinement was not too interesting.
sequently done this very thing.
Dornberger-Schiff (1962) has subThe structure proposed is much
like one the author picked in a study of the octahedral cation sheet
in wollastonite.
The pseudowollastonite structure consists of layers of
Ca octahedrally coordinated by oxygens.
These octahedral layers
are connected by rings of Si tetrahedra which share oxygens with
the Ca octahedra in'layers above and below the rings.
Dornberger-
Schiff (1962) reports the presence of diffuse reflections and radiation
streaks in pseudowollastonite diffraction patterns which are indicative of disregistry between adjoining octahedral sheets.
The phase equilibria between pseudowollastonite and wollastonite are of interest.
Pseudowollastonite,
thesized by Bourgeois (1882),
which was first syn-
can be formed by heating CaO and
Si.O2 above 12000 C. (melting point of pseudowollastonite: 15400 C.)
and will persist in a metastable state if cooled to room temperature.
It can also be synthesized hydrothermally at as
low as 6000 C. (Flint,
McMurdie, and Wells,
1938).
Although wollastonite can be formed
from pseudowollastonite under certain circumstances (Allen,
White,
and Wright, 1906),
no evidence has been presented that this takes
place in natural systems.
primary mineral.
Instead, wollastonite seems to form as a
It is still probable, however, that whether
wollastonite or parawollastonite occurs may have some connection
with the temperature of formation, even if pseudowollastonite is not
involved.
Hydrated calcium silicates.
Many hydrated calcium silicates
are known, most of which show some relationship with wollastonite,
such as a transformation to wollastonite upon being heated.
None
of these have been shown to have a wollastonite-like silicate chain,
however.
Not too much work has been done in this area and some
phases may have the wollastonite structure with hydrogen bonding
as in pectolite, or with H 20 in the large holes which exist in the
wollastonite structure.
There is one hydrated calcium silicate which bears mentioning
here, and that is xonotolite, Ca
6
60 (OH)2.
6 1 7 (OH 2 .
Mamedov and Belov
aeo'n
eo
(1955) have published a structure for xonotolite which seems to be
correct (Raw 20%).
This structure consists of silicate chains which
are similar to those in wollastonite except that the chains are paired
with the Si.3 tetrahedra in each pair sharing a corner.
Although no
investigation of xonotolite has been made in connection with this
121
thesis, there are certain aspects of the proposed structure which
should be further studied.
The primary reason for suspicion is the
great similarity between the reported diffraction diagrams for
xonotolite and those for parawollastonite.
For example, a in both
xonotolite and parawollastonite is approximately double the
wollastonite a.
Dent and Taylor (1956) report that the odd layers
photographs of xonotolite are streaked and very weak.
It could be
that xonotolite does not have a really different structure from
wollastonite and varies only in the way that the cells are put together.
Whether this is true could probably be determined by computing
interatomic distances from coordinates given by Mamedov and
Belov (1955).
Even if their structure is correct, there is no doubt
that it should be refined.
122
References
Allen, E. T.,
W. P. White, and F. E. Wright (1906), On wollastonite
and pseudowollastonite, polymorphic forms of calcium metasilicate.
Am. J. Sci.,
21,
89-108.
Bourgeois, L. (1882), Essai de production artificielle de wollastonite
et de meionite.
Bull Soc. Min. France, 5,
Dent, L. S. and H. F. W. Taylor (1956),
13-16.
The dehydration of xonotolite,
Acta Cryst, 9, 1002-1004.
Dornberger-Schiff, K. (1962),
The symmetry and structure of
strontium germanate, Sr(GeO3),
a-wollastonite, Ca(SiO 3 ).
as a structure model for
Soviet Physics- -Crystallography,
6, 694-700.
Flint, E. P., H. F. McMurdie, and L. S. Wells (1938),
Formation of
hydrated calcium silicates at elevated temperatures and pressures.
Bur. Stand. J. Res.,
Ginetti, Y. (1954),
21,' 617-638.
Structure cristalline du metagermanate de cuivre.
Bull. Soc. Chim. Belg.,
Zur strukturbestimrnmung von strontium-
Hilmer, Waltraud (1958),
germanat SrGeO
63, 209-216.
Naturwiss.
10, 238-239.
Lazaref, A. N., T. F. Tenisheva, and R. G. Grebenshchikov (1962),
The structure of barium silicates.
Soviet Physics--Doklady,
6, 847-850.
Bemerkungen zur systematik der
Liebau, Friedrich (1956),
kr istall strukturen von s ilikaten mit hochkondens ierten anionen.
Z. Phys. Chemie.,
206, 73-92.
Liebau, Friedrich (1959), Uber die kristallstruktur des pyroxmangits
(Mn, Fe, Ca, Mg)Si0
3
Acta Cryst.,
12, 177-181.
Liebau, F.
(1960),
Zur kristallchemie der silikate, germanate und
fluoberyllate des formeltyps ABX
.
N. Jb. Miner., Abh.,
1209-1222.
Mamedov, Kh.C.,
xonotolite,
and N.V.Belov (1955),
DokladyAkad. Nauk SSSR,
Crystal structure of
104, 615-618.
94,
Chapter VIII
Crystal structure of larsenite, PbZnSiO
4
Experimental work
Although the structure of larsenite has no direct connection
to those of wollastonite and pectolite, this account is presented
because many of the experimental and computing techniques used
by the author for wollastonite and pectolite were developed in the
study of larsenite.
As the structure of larsenite has not been
completely solved, this chapter represents more of a progress
report than a finished paper.
The solution of the larsenite structure is especially interesting
in that it appears that it will be a structure unlike that of any of the
olivines or other orthosilicates.
This, however, will not be certain
until the work is finished.
Occurrence.
Larsenite has been reported only from
Franklin, New Jersey.
The specimens are found in veins cutting
massive willemite-franklinite ore along with garnet, hodgkinsonite,
calcite, zincite, willemite, and clinohedrite.
The crystals are slender
needles 10 to 20 times longer than they are thick.
nated crystals have been found (Palache, 1935),
be doubly terminated.
Although termi-
none are known to
Larsenite has one good cleavage parallel
to (120) of the morphological unit.
I
RNEEMEMM
1
Unit cell and space group.
The unit cell and space group of
larsenite was published by Layman (1957).
Table 1 compares
Laymanrs cell data with that obtained in the present investigation
and with forsterite.
Layman stated that the larsenite cell apparently
was that of ollvine doubled along a and b.
As will be shown below,
this seems to be a fortuitous comparison.
The space group given
by Layman, Pnam or Pna, was confirmed by the author.
Since no
doubly terminated crystals have been found, it is difficult to establish to which of these space groups larsenite belongs.
Because,
however, of the large number of minerals reported in space group
Pnam versus only one in Pna (Donnay and Nowacki,
1954), Pnam
seems to be the more likely.
Intens ity collect Lon.
-1
(p = 911 cm for CuK
Since absorption is quite large
radiation) in larsentte, a thin needle was
selected for intensity determination.
The absorption correction
was made by assuming that the crystal approximated a cylinder
with a radius of 0.0015 cm (15 microns), giving a
IR of 1. 37.
The intensities were collected with a Weissenberg Geiger counter
diffractometer (Buerger, 1960) using CuK
radiation.
Care was
taken that the linearity limits of the Geiger tube was not exceeded.
The integrated intensities were obtained by measuring with a
planimeter the peak areas above background as recorded on a strip
chart recorder.
In all, 822 intensities were recorded.
1 26
Solution of the structure. The three-dimensi onal Patterson
function was computed using the Fourier program MIFRI1 (Sly and
Shoemaker, 1960).
At this point it was thought that the structure
might be solved by inspection if it were similar to that of olivine.
Table 2 gives the equipoint distribution of the atoms in forsterite
(Belov and Belova, 1951) and the possible equipoints in Pnam for
lar senite.
Examination of the Patterson sections normal to c shows
that all of the strong peaks lie on p = 0 and z = 1.
It is relatively
easy to determine whether Pb or Zn are located on centers of
symmetry in4 a or 4 b.
If 4a and/or 4 b were filled with Pb or Zn,
large peaks would occur in the Patterson at 00 -.
Since there is a
negative region at this point, 4a and 4b certainly do not contain
Pb or Zn and probably not 0.
Si is excluded since it never is found
on centers of symmetry in orthosilicates.
This leaves only 4 c and
8 d to be filled.
Since all the strong peaks in the Patterson lie on sections
which are separated by ?z it is probable that the heavy atoms are
in 4 c (the mirror planes at 4z and 3z).
peaks for Pb
(.082, .465,
With this in mind, inversion
and Pb2 with supporting satellites were located at
,) and (. 304, .118,
-),
respectively.
inversion peaks it was possible to construct an
function of the crystal structure.
From these
Pbl+Pb?
M 8(XY)
This map is given in Fig. 1.
The
-)'7
f
Table I
Comparison of larsenite and forsterite cells
Larsenite 1
Larsenite 2
X
For ster ite 3
a
8.23+.01
b
18.94+.01
c
5.06+.01
5.06
5. 997
z
8
8
4
formula
space group
3
19.1
PbZnSiO 4
PbZnSLO 4
Pnam
Pnam
Layman (1957)
2 Prewitt
8. 30 X
(this work)
Belov and Belova (1951)
4.765
10.23
Mg 2 Si
O
Pbnm
A
-128
Table 2
The forsterite structure (Belov and Belova, 1951)
4Mg 2 Sio 4 , space group Pbnm
Atom
Equipoint
Symmetry
Mg 1
4a
i
Mg
4c
m
Si
4c
m
O1
4c
m
4c
8d
2
02
03
-. 01
. 218
.415
.095
1/4
229
. 083
1/4
m
. 221
. 430
1/4
I
.262
.152
.027
-.
The larsenite structure
8PbZnSiO , space group Pnam*
Equipoint
Equivalent positions
Symmetry
4a
-0;
000; 00-;
2'
4b
100;
4c
xy 4;
xy;
8d
xyz;
2+x
22
2' 2
O1A ; 0-10; 01
-X,
2.-y,
+Y,4
Z1 z
x,
+y, z; xyz; ] -x, -+y,
x, y,
-z;
+X, I-yz
-Y,4
Y, -Z+z;
j-x,
*Equivalent positions for this orientation are not given in
International Tables, Vol. I.
2; +x,
+z;
Figure 1
Pb +Pb
2
M 8 (xy ) map of larsenite
Iuz
!SSq
zqqd
q
0
13j
large peaks are Pb and the somewhat smaller ones correspond to Zn.
It is almost certain that Si also lies on the mirror plane since the
short c axis would require that if Si were not on the mirror plane,
the silicate tetrahedra would have to be linked in chains across the
mirror planes and this cannot be the case in orthosilicates.
ing this, the most likely location for Si is at (.24, .44,
(.28, * 67,
Accept-
-) and
).
Although the cations are located rather easily, the oxygen
positions present a problem.
There is no information on the
M 8 (xy ) function which will give a consistent set of oxygen coordinates.
An attempt to find these locations is discussed in the next section.
Refinement.
A number of cycles of refinement were tried,
using a computer program written by Prewitt (Appendix I).
lowest R-factor obtained was 19. 7%.
refinement are given in Table 3.
The
The final coordinates for this
It is evident that the oxygens are
not located properly here since other refinement cycles in which
the oxygen coordinates were completely different gave R-factors
almost as low as 19. 7%.
A three-dimensional difference map using
the coordinates of Table 3 was computed to assist in relocating the
oxygen atoms, but apparently either enough signs were wrong or
enough reflections were in error to produce a rather ambiguous
map.
Some changes were made according to a best guess of what
information the map contained, but further refinement was unsuccessful.
It is noteworthy that the difference map showed a depression
Table 3
Atom coordinates for larsenite
Atom
x
y
Pb
.037
.270
Pb 2
.148
.059
.498
.943
Zn
z
1
4
1
1
Zn 2
.674
.656
1
Si
.796
.824
Si 2
0*
*
.226
.423
.988
.877
02
.649
.876
.352
.475
04
.091
.460
05
.250
.353
.427
06
.809
.775
.493
4
1
1
03
3
4
1
4
* The oxygen coordinates are provisional
'33
in the region of Zn2 which was not large enough to be the result of
misplacing one of the zinc atoms, but which indicates that something
is wrong with the proposed structure.
Discussion of results
Since the least-squares refinement has reached a point where
some modification of the larsenite structure has to be made, it is
useful to review the facts and to recommend what procedures
should be followed to reach a successful conclusion.
It is believed
that the answer will be forthcoming if a reasonable effort is put into
the solution.
Oxygen packing. One of the first things that should be considered is whether the oxygen might have an olivine-like packing
and consequently similar cation coordination, or whether the structure
might be related to that of willemite, Zn2S.Og, where the Zn is in
tetrahedral coordination.
The space group of forsterite and
larsenite are the same, but attempts to fit the Pb and Zn of
larsenite into the forsterite oxygen structure have not been successful.
It also has not been possible to find a reasonable structure
where the Zn are tetrahedrally coordinated.
The tentative inter-
pretation of this is that the larsenite structure is really a new one.
Other chemical and physical evidence should be collected
and tabulated before any further work is done.
This includes
refractivities,
interatomic distances, and coordinations of Pb and Zn.
One difficulty here is that there is very little reliable structural data
available on Pb and Zn silicates.
Because of this, the solution of
larsenite becomes even more interesting and points the way to a
much larger field of investigation.
Substructures and heavy atoms. Another approach to structure
solution which comes to mind is the heavy atom method.
Since,
however, there are two crystallographically distinct Pb atoms in
larsenite, this is not a foolproof approach because the structure
factor amplitudes due to Pb cancel for many of the reflections.
This
technique was used to determine the signs of about one-fourth of the
reflections after the lead positions were determined.
This is
discussed further below in connection with direct methods.
Although the cations lie on mirror planes at z = I and3,
and
the oxygens probably lie on both the mirror plane and in the general
position, there is no set of reflections which can be picked that
contain contributions from only part of the atom population.
The
trigonometric formula for the general position in Pnam,
h+k+l
A = 8 cos27r(hx- h-)
shows that for atoms with z =
of the value of 1.
h+k
cos 27r(ky+ -~)
1
cos27r(lz+'),
, the third cosine is +1 regardless
It is true that oxygen in the general position would
contribute more heavily to the even levels, and an electron density
calculation using only these reflections may be computed.
Direct methods.
The most interesting and promising method
for solving larsenite is with Harker-Kasper inequalities or through
suitable application of sign relationships (Woolfson, 1961).
As was
stated above, the signs of about one-fourth or 200 reflections are
known because if
F Pb+Zn>8f
then the sign of F Pb+Z
factor.
+ 32f ,
must control the sign of the whole structure
Tests have shown that within these 200 strong reflections,
the sign relationship,
s (H) s (Ht) s (H+H)
is always obeyed.
~ +1,
This relation and others can be used to find
signs for many other reflections.
136
References
Belov, N. V., E. N. Belova, N. H. Andranova, and P. F. Smirnova (1951),
Determination of the parameters in the olivine (forsterite) structure
with the harmonic three-dimensional synthesis.
Nauk. SSSR,
Doklady Akad.
81, 399-402.
Buerger, Martin J. (1960),
Crystal structure analysis. John Wiley
and Sons, New York.
Donnay, J. D. H. and werner Nowacki (1954),
Crystal data.
Geol. Soc.
Am. Memoir 60.
Layman, Fredric G. (1957),
PbZnSiO .0
Unit cell and space group of larsenite,
Am. Mineral.,
Palache, Charles (1935),
42,
The minerals of Franklin and Sterling Hill,
Sussex County, New Jersey.
Sly, William G.,
and David P.
U. S. G. S. Prof. Paper, 180, 80-81.
Shoemaker (1960),
MIFRI1: Two- and
three-dimensional crystallographic Fourier summation program
for the IBM 704 computer.
Woolfson, M. M. (1961),
Unpublished report.
Direct methods in crystallography.
Clarendon Press, Oxford.
'37
Chapter IX
Automatic crystal structure analysis
Because of extremely rapid developments in the fields of
automation, control, and digital computing, it is possible today to
design a system for nearly automatic crystal structure determination
and refinement.
There are some tremendous advantages to such a
setup, both philosophical and practical, and also some disadvantages.
The advantages are that a great deal of useful, reliable structural
information can be gained in a relatively short time and investigators
can begin to spend more time interpreting results than in repeating
the experimental details over and over again.
One disadvantage is
that the costs per unit time will be quite high even though the cost
per crystal structure should be much lower than at present.
Another disadvantage is that a few people will have to spend considerable time in setting up an automated system and in eliminating the
deficiencies that are bound to be present.
Present capabilities. Considerable progress toward largescale automation has been made in connection with this thesis, and
also other theses by Burnham (1961) and Peacor (1962).
Burnham
outlined a series of data reduction programs for the IBM 7090
computer which have since been augmented and improved and which
are listed below.
In addition, the automatic diffractometer is now
a reality and should be available soon in a number of forms.
The
13'8
author has done considerable work on possible control systems for
the Weissenberg diffractometer,
including the writing of the
diffractometer settings program described in Appendix III.
The following IBM 7090 programs are now in use in the
Crystallographic Laboratory at M. 1. T. and, with revisions, could
be incorporated into an automated system.
i.
Least-squares refinement of lattice constants (Burnham)
2.
We is senberg diffractometer settings (Prewitt)
3.
Data reduction and prismatic absorption correction (Burnham)
4.
Cylindrical and spherical absorption correction (Prewitt)
5.
Fourier synthesis (Sly, Shoemaker andYan den Hende)
6.
Structure factors and full-matrix least-squares
refinement (Prewitt)
7.
Structure factors and diagonal matrix least-squares
refinement (Van den Hende)
8.
Drilling coordinates for crystal structure models
(Peacor and Prewitt)
This program is now under development.
General absorption correction (Wuensch and Prewitt)
Future capabilities. These programs need to, be obtained or
written before an automated system would be practical.
1.
Diffractometer control
2.
Processing of diffractometer output
139
3.
Structure determination
(a) Minimum function
(b) Inequalities
(c)
Statistics
4.
Refinement - differential synthesis
5.
Output
(a)
Fourier plot - cathode ray tube
(b) Fourier plot - mechanical or electrostatic
(c)
6.
Structure drawing - mechanical or electrostatic
Interatomic distances and interbond angles
With an automatic diffractometer and the above programs
working correctly, a structure analysis could be practically
automatic except for selection, mounting, and cell determination
of the crystal and the writing of the final paper.
Some work would
be required in the supervision of the equipment and in transferring
data between the diffractometer and the computer, but this would
be very small in comparison with what is now being done.
Of
course, many experimental difficulties can be expected before
such a system is perfected, but it is not inconceivable that, once a
typical mineral crystal is set up, the whole process from data
collection to final refined structure and illustrations could be
carried out in two weeks.
The time required at present for the
same thing can vary from a few months to over a year.
This does
not include protein or other structures with especially large cells,
or situations in which special problems arise.
Time required for
these latter investigations is often on the order of several years.
As
Expected results of automated crystal structure analysis.
was stated earlier, automation will allow the investigator much more
time in which to evaluate the results of his work--that is, unless
he spends the extra time working on more different structures.
In
either case, there will be opportunity to do more thinking about the
physical rather than the mechanical aspects of the problem.
As it
is today, the pressure for turning out results may cause important
concepts to be overlooked and poorer papers to be written.
Another, less obvious, result of automation is that the data
and therefore the results will be much more reliable.
Although the
less expensive human operator can set an instrument as well as a
machine for any individual diffraction intensity, he cannot possibly
be as consistent over several hundreds or thousands of observations.
Experience has shown that data from manually operated diffractometers
have contained errors which could have caused the more difficult
problem to fail completely.
Automation will enable the investigator to obtain reliable
information on many related compounds such as different pyroxenes
which would have been skipped previously as not being a good
investment of time.
From this work, much information about such
14-
topics as bonding, polymorphism, and phase equilibria should be
obtained, thus enabling the crystallographer to become more involved
in what he is working on than in what he is working with.
142
Part C
Appendix I
SFLSQ3,
a structure factor and least-squares refinement program
for the IBM 7090 digital computer
Introduction
The writing of this program was begun in the spring of 1961,
a few months after the installation of an IBM 709 digital computer at
the M. I. T. Computation Center.
At that time this laboratory was
using the crystallographic least-squares program, ORXLS, written
by Busing and Levy (1959) for the IBM 704.
Since many difficulties
were encountered in using the 704 program on the 709 and since the
running times were so long that many problems could not be undertaken, the author decided to write a program modeled after the one
by Busing and Levy which would be very flexible, faster, and
operable under the Fortran Monitor System.
The resulting program,
SFLSQ3, has been in use for over a year and has been used successfully by a number of people for both structure factor and least-squares
calculations.
The programming of SFLSQ3 is in FORTRAN with the
exception of the normal equation matrix storage routine, the Busing
and Levy matrix inverter, and library routines, which are written
in FAP.
Much of the program is broken up into subroutines which
communicate With each other and with the main program through a
large common storage.
This enables the user to make many changes
which would be impossible with most other similar programs.
It is
felt that by coding in FORTRAN and by providing a detailed writeup
and references, the user should find this program no harder to set
up and run than others of similar type which have been distributed.
Although the mathematical method in SFLSQ3 is similar in
many respects to ORXLS of Busing and Levy (1959),
the programming
is completely new except for the matrix inversion routine ORSMI
(Busing and Levy, 1962) which was obtained from Dr. Busing through
Dr. Karl Fischer.
As SFLSQ3 is now constituted, the full normal
equation matrix is inverted to obtain the necessary parameter shifts.
It would be quite possible to provide the option of refining using only
the dLagonal matrix, a feature which would considerably reduce the
time required to run the program.
Then, presumably, there would
be an optimum procedure for carrying out a complete least-squares
refinement, depending on the number of observations and the number
of variable parameters.
For example,
in a problem involving a
large number of observations and variable parameters, considerable
time might be saved by first refining using the diagonal approximation
until convergence is almost obtained and then switching to a few
cycles of full matrix and anisotropic temperature refinement, so
that as good a solution as possible is obtained.
For smaller problems,
the most advantageous approach would probably be to use the full
matrix in all the computations.
When run under the Fortran Monitor System (FMS),
SFLSQ3
consists of a main program and 16 subroutines, plus subroutines
called from the library tape.
The program as distributed will
refine up to 172 variables (full matrix refinement) and an essentially
unlimited number of reflections.
It was originally intended that
subroutine CHLNK3 could be compiled as a main program so that it
would be read in separately as a CHAIN LINK, thereby increasing
the number of registers available for storing the matrix.
This can
still be done, but it might be more useful to consider the IBM 1301
Disk File which will soon be available as the means for storing
larger matrices.
Common storage .
Communication between the main program
and the subroutines is principally through a large block of COMMON
storage which is divided into six dummy arrays, CMA, CMB, CMC,
CMD, CME, and CMF.
Variable symbols are assigned to portions
of these arrays by means of EQUIVALENCE statements.
Table 1
is a list of the symbols which refer to COMMON storage.
Sense indicators.
The array CMA contains the sense indica-
tors which control much of the flow of the program.
The definitions
of these indicators are given in the section on running the program.
They will be referred to frequently in the sections that follow.
Tape usage.
given in Table 2.
The tape assignments for this program are
A copy of the M. I. T. IOU subprogram which
146
Table 1
SymbolLs assigned to COMMON storage
Symbol
Di.mension
CMA
50
CMB5
07
CMC
6 51
CMD
7 00
CME
4 19
CMF
50
AR
Dummy array
t
149 69
V
It
Normal equation matrix
2 00
TITLE
Function
Vector matrix
12
Title for printed output
MODE, INV,
ISAN, NPU, NEW
NU, NS, NFOUR,
NCOR, SUMDL,
KARD, ID, NFSQ
See directions for preparing sense
card.
A, B, C, ALPHA,
BETA, GAMMA
each
Cell constants
SUMDL
1(4w A) 2
hkI
NPAR
Total number of parameters
NVAR
1.
NCOUNT
KSEL
S
BO
5 04
50
1
Number of parameters varied
Number of observations used in
refinement
Parameter selection words
Scale factors
Overall temperature factor
~~~-'qm
-~
Table 1 continued
MF
50
Form factor indicator
50 each
Atom coordinates
BI
50
Isotropic temperature factor
BI1, B22, B33,
B12, Bi3, B23
50 each
Anisotropic temperature factors
SYM
50
Atom "scale factor" or multiplicity
NAME
50
Atom name for printed output
X, Y,
Z
NF
FORM
F
1
32, 20
Number of form factor tables
Form factor tables
20
f.
L
ASTER
4
Asterisks for printed output
RHO, RHOSQ
I each
s nO/>A-,
ARG
sin q/2
Arguments for form factor tables
32
ISAVE
Indicator of last entry into form
factor table arguments
FOBS
F
FCAL
F c(hkl)
Fc
AOBS, ACAL,
BOBS, BCAL
(hkl)
A 0, A c,
o
c
Bo, B c
DELTA
(F
F ) =A
c
SIGMA
Observation error
EXTI, EXT2
1each
1 each
o
- S
m
Optional extra input
SQRTW
MSI.
Scale factor subscript
r
Ln.
Table 1 continued
MREJ
1
Rejection indicator
MH, MK, ML
1 each
Fixed point h, k, 1
TH, TK, TL
I each
Floating point h, k, 1
DA, DB
Derivatives of A
200 each
SFC
1
S
SMSFC
1
2 S F (hkl)
hkl m c
SQDL
1
(wA)
DELTAI
1
SUMFOI
1
WDLI
1
(hkl(wA)
WFOI
1
(
DELTA2
1
SUMFO2
1
WDL2
1
WFO2
1
m
hkl
and B
c
c
F (hkl)
c
2
A
including zero F Is
o
- F including zero F Is
o
hkl o
2
including zero Fo Is
(wF))2
0
hkl
including zero F Is
o
Same as for preceding four symbols
except without zero F Is
Table 2
Tape assignments for SFLSQ3 at M. I. T. Computation Center
Use
Logical
Physical
1
Al
Library
4
A2
Input
2
A3
Output
8
A4
Derivatives if CHAIN JOB is used
9
A5
Optional input for ERFR2
(Fourier synthesis)
5
BI
6
B2
Chain tape (optional)
7
B3
Scratch
3
B4
Punched output
10
B5
Binary output tape used as input
to next cycle
correlates the logical and physical tape designations is provided with
the program deck.
If the physical designations do not correspond
with the usage of a particular computing center,
it is generally easier
to change the IOU rather than the logical tape numbers in the
FORTRAN input-output statements.
One should, however, check with
the computing center if he is uncertain as to how to proceed.
Punched output. At the M. I. T. Computation Center, punched
output may be produced from a tape or on-line depending on whether
sense switch 4 is down.
The FORTRAN statements, PUNCH N, List,
in subprograms PRSF and CARD may have to be replaced by WRITE
OUTPUT TAPE M, N, List, if punched output is desired at a particular center where the sense switch option is not available.
Otherwise,
all card punching will be done on-line.
A subprogram from the M.I. T. library, MIFLIP, is
provided with the program which will cause identification cards to
be punched along with any off-line punched output.
If this feature is
not desired, a dummy subprogram, SUBROUTINE PILFI (FLP, N),
may be compiled or statements 16 and 17 in READI removed.
Description of the main program and its subroutines
MAIN. Although the main program is primarily used as a
means of connecting the various subroutines,
essential computational functions.
it does perform some
For example, the statements
from 500 to 501 convert temperature factors from isotropic to
anisotropic form when so requested by the sense card.
At state-
ment 8, the program tests whether reflection data is to be read from
A2 or B5 and then reads and processes one reflection at a time until
all are read in.
or read from B5.
For each reflection, the value of sin
/X
is computed
When this quantity is computed, the program
evaluates
2
2
p
=
2
2 *2
(sin 6/X ) = (h a
2 *2
+k b
2 *2
+1
c
**
**
+2hka b +2klb c +21hc
**
a )/4. (1)
The main program also computes the derivatives of F (hkl) or
-c --
2
F (hkl) with respect to varied parameters beginning at statement 38.
--
These relations are given in Table 3.
In order to be able to run consecutive structure factor
calculations or cycles of refinement with a minimum of difficulty,
all input data with the exception of title and sense cards is written
on B5 at the end of a structure factor or least-squares calculation.
The main program (or subroutine CHAIN) has to put some of this
information fi rst on B3 so that B5 can be updated on each cycle.
Table 3
Derivatives of F (hkl) or F 2 (hkl) with respect to varied parameters (After Busing and Levy, 1959)
C
C
F (hid)
c
Computed
Quantity
Centrosymmetric
s
F
m c
2 s
8
(s mF )
(
as
m
m
8(s
F
8B
m
Sp
s m exp(-B o p2 ) (Ac2 + Bc 2)
exp(-B p ) Ac
o
c
m
m
2
c
2 s
m
c
m
2
-p)-p
m c
0
(sm
Non-centrosymmetric
2
Ac
exp(-B p )Ac
o
(sF
)
m c
F
(s
Bp
s
m
2
2
exp(-Bp ) (Ac + Bc
o0p8
Ac(
8Ac
8B
+ Bc(--)]
Table 3 continued
F C2(hid)
Computed
Quantity
(s
Fc)
2
O(s m F c )2
Os
4 sm
2
F
m
8B 0
2
2
2
exp(-2B0p ) Ac
s F )2 /s
m
m
8(s
O(s
Centrosymmetric
c
Non-centrosymmetric
s
2
2 (s
m
2
2
2
exp(-2B0p ) (Ac + Bc )
m
F ) 2 /s
c
m
)
c
F )
m c
Op
- 2 p 2(s
8 s
m
F )2
m c
2 exp(-2B p
o
-2p2(s
O2)A
8Ac
8p
2 s
m
m
F )2
c
2 exp(-2B p2) [Ac(Ac)
+ Bc(8Bc
0p
Op
o
I.
Nor-
The following symbols not listed in Table i are used in the
main program.
Symbol
Dimension
DF
200
Function
Storage for derivatives
ALPHA 1, BETAI,
GAMMA I
1 each
Reciprocal cell angles in
radians
AA, BB, CC, ABC,
BCA, CAB
1 each
Functions of cell paramete r s
I,J, K
1 each
Variable subscripts
N, NT
I each
Number of storage registe rs
in normal equation matrix
P1
T
RAD
7r/180.0
TO
Exponential of B
C I, C 2, Q
Terms used in calculating
derivatives
LAST
Indicator of last reflection
card
IEND
Indicator of the end of run
MS
Scale factor subscript
WDEL
V w-A
READ 1.
0
The title, sense, and cell cards are read from A2.
If data is available from a previous calculation, all but the reflection
data is read in from B5.
PILF1 is called, if requested, to write
identification of off-line punched output on B4.
The only symbol defined in READ I is FLP, an array of four
registers used for storing the punched output identification.
READ 2. Reads and stores form factor tables unless the fis
were read from B5 and are not to be altered.
READ 3.
Reads and stores scale factors, overall temperature
factor, atom coordinates and temperature factors, atom multiplicities,
atom names, number of parameters,
and parameter selection card.
If this data was read in from B5 in READ I and is to be used without
change, the program does not attempt to read A2.
SET. See STMAT.
LOOKUP. If f's have not already been computed in a previous
calculation, this routine performs a linear interpolation in the form
factor tables.
The program "looks" both ways in a table of arguments
(array ARG) to find the location of sinD/A for a particular hkl and
then computes and stores interpolated frs for all different types of
atoms in array F.
Each succeeding entry to the argument table will
be at the point of last exit so that the program is more efficient if
reflections wi th similar sin 0/X are grouped together.
It should be noted that it would be relatively easy to change
the interpolation scheme to a more sophisticated one or to substitute
a subroutine which would compute the form factors directly.
The
only requirement is that the fIs are stored in array F in the proper
order.
WEIGHT.
either reading
Weighting for least-squares is accomplished by
(fw
or in this subroutine.
dummy subroutine.
= -)
from reflection data cards or by calculating
If U is read from cards, WEIGHT must be a
The main program computes V-w = 0 if a = 0.
1 56
If a weighting scheme requires the calculated structure factor,
vW
could be computed in REJECT.
Possible weighting schemes are given in International Tables
(1959), Vol. II, p.
3 28,
Cruickshank (1961), Evans (1961),
and
Burnham and Buerger (1961).
SPGRP.
Since this is the only subroutine which will require
alterations for most routine work, considerable space will be
devoted to its details.
Many arguments have been advanced for
coding space group manipulations in one way or another.
In order
to avoid this problem and because there are definite advantages in
having particular space group routines for specific applications,
the choice is left up to the user,
The main program requires that
SPGRP computeACAL and, if the space group is non-centrosymmetric, BCAL.
It also requires computation of derivatives of ACAL and
BCAL with respect to the parameters that are to be varied.
Two
space group routines, however, are provided with the program.
One is a general routine which can be easily adapted for any space
group and with which anisotropic temperature factors can be refined.
The other is written for space group P. only and is representative
of routines which can be written for other specific space groups by
using the Lonsdale formulas for the general position (International
Tables, 1952, Vol. I).
Unfortunately, except for P1 and P!,
not easy to refine anisotropic temperature factors using this
approach.
it is
The remaining part of this section will be used to describe the
two subroutines distributed with the program.
Those interested in
other approaches to the problem are referred to papers by Rollett
and Davies (1955),
Marsh (1961),
Trueblood (1956),
Cruickshank (1961),
Hybl and
and International Tables (1959), Vol. II, p. 326-328.
Table 4 gives the quantities which are calculated by the
general space group routine.
reflection.
SPGRP is entered once for each
A (hkl) and B (hkl) are computed and stored in ACAL and
-c ---
BCAL respectively, while derivatives of A and B are stored in
-c
-c
arrays DA and DB, respectively.
The only changes which must be
made to allow for different space groups are in the 300 group of
statements unless modifications for atoms in special positions are
required.
The general space group routine computes the contribution to
the structure factor of each atom in the cell exclusive of atoms
related by translations or centers of symmetry.
For these latter
groups of atoms, the contribution of one atom is computed and
multiplied by a multiplicity factor M (designated SYM in the program).
The transformation of symmetry related atoms is accomplished by
transforming indices instead of atom coordinates.
This is done
mainly so that anisotropic temperature factors can be handled
properly.
The symmetry considerations are resolved by supplying the
routine with the transformed indices.
As an example, consider
Table 4
Quantities evaluated by general space group subroutine
j
Atoms i are transformed through
Computed
Quantity
Isotropic temperature factor
Ac(hkl)
M. f. exp(-T.)
L
L
Anisotropic temperature factor
cos27r(h.x.+k.y.+1.z.)
J L
L
J L
J
M f.
L
L
L
I
j
exp(-T..) cos2r(h.x.+k.y.+1.z.)
M. f. exp(- T.)
L L
L
s in2(h.x.+k.y.+. z.)
L
M. f.
L
L
Ji
Lj
j
iL
Bc(hkl)
equivalent positions
L
JL
)
L
L
exp(-T..) sin27r(h.x.+k.y.+1.z.)
L
L
LLJ
B~ 2
B~ph.
Tij
T.j
j
e
Ac
M
A
E M.
LJ
2
2
*+k. p .+1. p .+2h.k.1
ii L j Co
2 Ls
33 L
J j
p
f
cos..
f. exp
L L
L
2
L
exp.. cos..
L3
L
8jBc
0fBc
8 M.
L
ex.
L
-c2rM.
8
aX._
x;
L
si
nM.
L
Lj
Ac i
f. exp
h.cs..
xLpL h
2_M.fep.hc
LL
J
Lco
L
exp.. sin..
3
L3
-27rM. f.
h
Li
s.__
L
.
.
L
h.ex
j
i.
x.
L
.
L
3
co
.
LiJ-
~J13
.+2h.l.p3 .+2k.l.p2.
L
J j iSL
23 L
Table 4 continued
Computed
Quantity
Isotropic temperature factor
2
8 Ac
B
Cos..
s..
p M. f. exp.
L
8 B.L
L L
8 Bc
2
-p
8B.
Anisotropic temperature factor
M. f. ex.
L
7
L3
SLn..
L
8 Ac
a %j
i
.Bc
a P
- M. f.
h. exp.. cos..
- M. f.
h. exp.. s in..
L
.
3
p = s in 9/A
M.
B.
L
L
is atom multiplicity
is the isotropic temperature factor
Bc(hkl) is the imaginary component of the structure factor
L3
L3
L3
L3
r);
space group Pnam.
The general position in Pnam is represented by
the coordinates
xyz; L+x,
-y
;iy
plus their centrosymmetric equivalents.
z
-x,
zy
It is evident that
cos 27(htx+kty+lz+t) = cos 27r(h(*+x)+k( -y)+l( -z))
t
where hl = h, k
= -k, 11 = -1,
and t = (h+k+1)/2.
and I are stored in TH, TK, and TL, and hi, k T , 1,
RL, and T.
(2)
In SPGRP, h, k,
and t in RH, RK,
The only requirement for Pnam is that RH, RK, RL, and
T be transformed to correspond to each of the four equivalent
This is accomplished by writing the following
positions above.
sequence of FORTRAN instructions and compiling them into SPGRP.
Note that only those indices which are different from the previous
loop have to be computed.
300
DO 413 J=1, 4
301
GO TO (302, 303, 304, 305), J
302
RH = TH
RK = TK
RL = TL
T
= 0.0
GO TO 400
303
RK =-TK
RL =-TL
T = (TH+TK+TL)/2.0
GO TO 400
161
304
RH=-TH
RL = TL
T = TL/2.0
GO TO 400
305
RK = TK
RL =-TL
T = (TH+ TK)/2.0
400
413
PHI = TPI*(RH*X(I)+RK*Y(I)+RL*Z(I)+T)
CONTINUE
This is the only change required for most situations.
Extra modifi-
cation may be necessary, however, when computing anisotropic
temperature factors for atoms in special positions, for atoms with
redundant coordinates, and for anomalous dispersion corrections.
Anisotropic temperature factor computations for atoms in
special positions of Pnam are not difficult because the only restraints
are that for atoms lying on the mirror plane in equipoint 4c,
P13
This can be taken care of by simply not
23 = 0 (Levy, 1956).
varying
p13
and
p2 3 .
When refining structures in space groups of
higher symmetry, however, the constraints on the anisotropic
temperature factor lead to special coding problems.
example, the space group P6
Levy (1956).
3
/mmc
Take, for
discussed in this regard by
He finds that for equipoint 12k with symmetry n,
p12
=
22,
13
23'
162
These restrictions would be handled in SPGRP in the following way.
SPGRP normally evaluates the anisotropic temperature factor as
given in Table 4.
Since P12 and
p31
are redundant, they must not
be varied.
The expression which is evaluated is then
exp{-[ h.p
.+(k +h.k.)p
j
iiL
J
J 3
22L
+.+1.
p l+(2h.k.+k.l.)P
J P33L
3 J
J J
23i
] }.
(3)
This would be accomplished by substituting the following statements
for those in SPGRP from 407 to 407 +5.
407
RHH = RH**2
RKK = RK**2+RH*RK
RLL = RLL**2
RHK = 0.0
RHL = 0.0
RKL = 2. 0*RH*RK+RK*RL
In the event that different types of special positions are present, a
statement at 407,
GO TO (Ni, N 2 ,... N7), I,
where I is number of the atom being considered, would direct the
program to the proper index transformation.
After a cycle of refinement is completed, the redundant pts
should be set to their proper values in TEST before the test for
negative temperature factors is made.
Redundant positional
stateparameters which are handled similarly in the 300 group of
ments can also be reset at this point.
163~
If any particular diffi culty is encountered in determining what
to do about ani.sotropic temperatures or atom coordinates in specific
space groups reference to Levy (1956),
(1961),
Busing and Levy (1959),
Trueblood (1956),
Cruickshank
or International Tables (1959),
Vol. II,
might be helpful.
Anomalous dispersion can easily be allowed for in any version
of SPGRP.
In the general space group routine, the dispersion terms
One way of doing this is
are added in the 500 series of statements.
to compute the contributions to the structure factor for the real and
imaginary scattering factor terms separately and then add the components of the structure factor vectorially.
For example,
if the total
scattering factor is
f = f
+ Af t +iAf",
(4)
then (neglecting temperature factors),
{(f +Af)cos
A =
B ={(f
and
c
J
sin4 = 0.
(5)
}
(6)
+Aft) sin4+Af"cos
for non-centrosymmetric space groups.
groups,
-Af"sin4}
In centrosymmetric space
For atoms in the cell which do not exhibit
appreciable anomalous scattering, Af = Af" = 0.
Since centro-
symmetric structures are treated as though they are non-centrosymmetric, the user must remember to run the program with INV = 2
and to adjust the multiplicity to account for the fact that the main
program will not multiply ACAL by two (see Table 3).
As an example,
b
164
the following sequence of statements is suggested for a problem with
two atoms, the first being an anomalous scatterer and the other not.
Data for the first atom, Fe in CuK
International Tables (1962),
radiation, is taken from
Vol. III.
500
GO TO (501, 502, 502), ISAN
501
TMP = EXPF(-BI(I)*RHOSQ)*SYM(l)
GO TO 503
502
TMP = SYM(I)
503
GO TO (504, 505), 1
504
TMPA = (F(JJ)-1. 1)*TMP
TMPB = 3.4*TMP
GO TO 506
505
TMPA = F(JJ)*TMP
TMPB = 0. 0
506
AR = TMPA*SMCP
Al = TMPB*SMCP
BR = TMPA*SMSP
B I = TMPB*SMSP
507
ACAL = ACAL +AR
-
BI
BCAL = BCAL+BR+AI
508
GO TO (700, 600), MODE
Statements 600 to 600 + 3 compute constants required for the
derivatives.
If no anomalous scattering correction is used, TMPB
should be made equivalent to TMPA with a statement, TMPB = TMPA,
or EQUIVALENCE (TMPA, TMPB).
Derivatives are calculated in the general SPGRP routine as
noted in Table 4.
For each atom in turn, the program computes
the contributions to A
-c
,
B
-c
,
and derivatives of A
-c
and -c
B
respect to the atom parameters that are to be varied.
with
If the number
of derivatives calculated exceeds the number of parameters to be
varied the program skips to the calculations for the next atom in the
list.
Space group calculations which evaluate the Lonsdale formulas
differ 6nly in that the expression for the general position is used
instead of finding the contribution of each symmetry equivalent atom
separately.
The computation time for this method is probably some-
what less than for the general routine, but there are two drawbacks.
One is that auisotropic temperature factors cannot be handled
conveniently except in P1 or P!, and the other is that the expressions
for the derivatives have to be worked out separately for each space
group.
An example for P1 coded in this way is supplied with the
program deck.
There are undoubtedly many variations in the methods of
calculation of structure factors which could be profitably used in
SPGRP.
Considerable machine time could be saved In spec ial
problems if the coding were done especially for those problems.
One obvious way to cut down on the running time would be to program
SPGRP in FAP.
It is estimated that the number of machine instructions
F
166
in the general SPGRP routine could be reduced by at least one-half if
the routine were written in FAP.
Symbols unique to the general SPGRP routine are listed below.
Symbol
Dimension
Function
Subscript for derivatives
DA and DB
Subscript for KSEL, the parameter selection words 27r
TPI
27r
PHI
4. = (h.x.+k.y.+l.z.)
-3- L
3J;-L -J3-L
CP
cos4. or exp. cost.
SP
s in.
SMCP
o r exp. sin.
I cos4.
1
SMSP
RH,RK, RL, T
L
Z sin.
3
3
h., k., I., t
~3 -3,h.2
-3
Exponential of anisotropic
temperature factor expression
-
RHH
E
CPH
2 h. cos.
TMP, TMPA, TMPB
Temporary storage
AR, A, BR, BI
Contribution of 1 atom to
structure factor
TA, TB, RA, RB
Derivative multipliers
REJECT. Any reflection not wanted in the refinement can be
rejected here by storing zero in MREJ.
Otherwise, all reflections
167
except those for which F
-c
refinement.
is identically zero will be included in the
The criterion for rejection can either be computed from
data available in COMMON or can be sensed from information
punched in the EXT I and/or EXT 2 fi elds of the reflection data cards.
EXT I and EXT 2 are stored in COMMON also.
STAT.
This subroutine accumulates statistical data for
R-factor calculations or any other desired reliability indicators.
Several registers in addition to those used in the distributed routine
are available in array CMF for transmitting this data to PRST, the
routine which prints statistical results after all structure factors
have been computed.
The quantity
(7)
A(hkl) = F o - s m F c
is given the sign of F
-c
F
is computed in this routine, where -o
.
This
differs from many refinement programs which compute the difference
F and F
of the absolute magnitudes of -o
-c
.
Of course, in non-
centrosymmetric structures, F-c is always positive.
The only symbol unique to STAT is SQFO which contains
(wA)2
PRSF.
This routine controls the structure factor output,
both printed and punched.
For each reflection the following is
written on the output tape:
h,k,l, IF o
, A , B
s F , A
1, -n-c
-o
-c
-o
,
B
-c
,
A,
A/,
asterisks where
i
A
_-o
=
IF
-o
(A /sF)
-
m
F(B /s
andB
F).
c -in-c
-o
B
-o
and B
6
-9
are not
-c
written on tape for centrosymmetric structures in SPGRP unless an
One asterisk is
anomalous dispersion correction has been made.
written if the reflection has been rejected from the refinement and
two are written if A/
> 2. 0.
Provision is made for punching of cards for input to ERFR2
(Sly, Shoemaker, and Van den Hende,
, B_), iF o-/s
h,k,1, A o , Ac , (B
-o -c
--
1962).
In this option,
I are either punched on-line or
written on B4 for off-line punching, depending on whether sense
switch 5 is up or down.
Since the option of on- or off-line punching
may not be available at a particular computing center, PUNCH
statements, 8 and 64, may haveto be changed.
This same information
may, however, be wri tten on A5 so that input to ERFR2 does not have
to go through the punched card state.
Enough information is available in COMMON to enable the
user to tailor the output to suit almost any requirement.
One might,
for example, substitute new SPGRP and PRSF routines which would
calculate and put out unitary structure factors.
Special symbols used in PRSF are listed below.
dimension of 1.
Symbol
Function
C I
F /(s
DS
Iiat
-o
FOSC
|IF I-m
IIndicator
F )
-m-
for a ster isks
All have a
169
STMAT and SET.
Since the storage of the normal equation and
vector matrices is the most time-consuming single operation in large
The entry point SET is
problems, this routine is written in FAP.
used to set up the indexing and addresses needed in STMAT.
The
storage loops are about as efficient as possible for this type of
program.
STMAT is entered once for each reflection accepted in the
refinement.
PRST.
The statistics accumulated in STAT are written on A3.
These include various R-factors and an error of fit given in Table
S.
Extra symbols used in PRST are as follows:
RI, R2, WRI, WR2
R-factors
RSUMDL
(E w
TEST
)2
Error of fit
CHLNK3.
Presently constituted as SUBROUTINE CHAIN.
It
may be compiled as a main program and run as a chain link with
SFLSQ3 which would then call the FMS subroutine CHAIN instead of
CHLNK3.
This requires several modifications in the main program
and STMAT as well as CHLNK3.
CHLNK3 calls the matrix inverter SMI, writes the Geller
correlation matrix (Geller,
1961) on A3,
computes and applies the
parameter shifts, and writes the refinement results on A3.
Following the notation of Busing and Levy (1959) and Geller
(1961),
Table 5 lists the various relations evaluated by CHLNK3.
Only the binary deck of the matrix inversion routine, ORSMI, is
supplied with this program.
However, a description of the numerical
methods used has been published by Busing and Levy (1962).
Three other subroutines are called by CHLNK3 which will be
discussed below.
One of these, TEST, tests whether the temperature
If the user so desires, the symbol K
factors are positive-definite.
in the calling sequence to TEST can be made non-zero if one or more
temperature factors are not positive-definite and the run will be
stopped regardless of how many cycles are left to be run.
zero, however,
If K is
the program will attempt to go through another cycle
if requested to do so.
If a singular matrix occurs, CHLNK3 will list the offending
elements on A3 and terminate the run.
A singular matrix is
generally the result of trying to vary parameters for which derivatives
have not been computed.
This can occur, however, when an attempt
is made to vary atoms in special positions or when varying c'oordinates
of overlapping atoms using two-dimensional data.
The following symbols not in COMMON are used in CHLNK3.
Symbol
ART
Function
Dimension
20
Line of correlation matrix
EP
1
Parameter change
ERFIT
1
FIT 1(-n)
FIT
1
Ii, 12, IT, MT, NV
I each
Subscripts used in generating
correlation matrix output
171
Table 5
Expressions evaluated in CHLNK3 (after Busing and Levy, 1959)
Quantity
Normal equation matrix
Relation
D7) (~wD )
(N
h,
aQ
Vector
v.
=
Inverse matrix
b
= A../d
( ~wD ) ( I- Aw
JL
Lj
Parameter shift
A p. =
3Db..v.
Error of fit
E
{
Standard error of Ap.
o-(p.)=
Correlation coefficient
pij
L
=
=
J
LJ J
V
LLL
(,w)2
m)
E
(/,)]
b iJ./[(1T.)
LL
JJ
Definitions
[ (sign of F ) F ]
c
o
A
s
m
scale factor
D.
derivatives
A.
cofactor of a..
d
det a
p.
parameters
L
-
s
m
F
c
IJD, U, NM, N, NT
1 each
Subscripts used in manipulating
least-squares matrices
I, J, K
1 each
Subscripts
KA
I
Positive-definite temperature
factor indicator
SUMCH
Ez
L
piv.
L
L L
TEST.
This subroutine is used to test whether the new
temperature factors are positive-definite and may also be used to
reset relations between parameters.
factor B
If the overall temperature
it is added to the individual atom temperature
is non-zero,
factors and the following tests are made.
For isotropic temperature factors,
.01 and K is left at zero.
if B < 0, B. will be set to
L
L
For anisotropic temperature factors the
Otherwise K will be non-zero
following relations must be satisfied.
and the program will stop upon return to CHLNK.
p .0,
lit-
p
>0
>0, p
3 3L22L7-
p1 1 p1 2 p1 3
p1 2 P22 p2 3
0
p1 3 p2 3 p3 3
23
>220
P23 P33
P11
13
P13 P33
> 0
11
12
>
0
p1 2 p2 2
The program can be changed so that after these tests are
made, the offending elements are altered so that they will pass the
positive-definite tests.
Another feature that could be added is to
'73
compute an equivalent isotropic B for each set of anisotropic pts.
See Hamilton (1959) or Lipscomb et al. (1956) for relations
between
B and Prs.
CARD.
This subroutine will write the new atom parameters
on B4 for off-line punching.
This is particularly useful if a large
number of atoms are being refined.
DIST. A subroutine, not written as yet, which will compute
interatomic distances and angles after the end of a cycle.
Running the program
Data deck.
directive below.
The data cards are prepared according to the
An understanding of FORTRAN format statements
is assumed.
FORMAT
1.
TITLE: Any identification in cols. 1-72
2.
FLP:
Information for identifying punched output.
May be blank if NPU below is zero.
The problem
and programmer numbers are for use at M. I. T.
Actually any Hollerith characters may be used.
Col.
1-6
Program number
7-12
Programmer number
13-24
Blank
4A6
-174
3.
Sense card.
Col.
MODE:
1 - structure factors only
II
2 - structure factors plus
least-squares
2
INV:
1 - centrosymmetric
I1
2 - non-centrosymmetri c
3
ISAN:
1 - isotropic temperature factors I
2 - anisotropic temperature
factor s
3 - change from isotropic to an-
isotropic before refining
4
NPU:
0 - no punched card structure
factor output
II
I - punched card structure factor
output (writes tape B4)
NEW(5): 0 for old values and I for new values
of:
5
5IH
(a) number of form factors and
cell constants
6
(b) form factor tables
7
(c) atom parameters
8
(d) key card and parameter
selection cards
9
(e) reflection data
This allows the user to select whether the old values from a
previous structure factor or least squares calculation (written
in binary on B5), or new values in the input deck are to be
used.
There are restrictions on the changing of certain of
these, e.g., the number of form factor tables may not be
altered without changing the card corresponding to (a) above.
These values must all be 1 unless the binary output tape (B5)
from a previous run is available to the program.
If data from
B5 is used, the corresponding cards must not be in the input
deck.
10-11
NU:
number of atoms in the asymmetric
unit
12
number of scale factors
12
12-13
NS:
14
NFOUR: 0 - no BCD output tape (A5) will be
I1
wr itten
I - BCD output tape (A5) will be
written with end of data card for
ERFR2 (1 in col. 1) and rewound
2 - BCD output tape with end of
data card and not rewound
This option permits writing a tape with the title above as the
first record and with h, k,1, A , A , and B
sary, plus IF0/s
and B , if neces-
i i n succeeding records for input to
ERFR2.
15
NCOR:
0 - no correlation matrix
11
1, - will write the Geller correlation
matrix on A3 for off-line printing
(Used only in mode 2)
16
KARD:
0 - no punched card output of atom
parameters
11
1 - the new scale factors, overall
temperature factor, and atom
coordinates and temperature
factors will be written on B4 for
off-line punching
17
ID:
0 - no interatomic distances and
interbond angles
11
176
1
- will compute and put out inter-
atoirnidcletances and interbond
angles after end of cycle.
Subrou-
tine DIST which computes these
functions is not available at this
ti me.
18
NFSQ:
I1
0 - refine on F
11
0
I - refi ne on F 2
0
4.
Cell card (note that angles are used and not cosines
as in ORXLS)
Col.
5.
1-2
Number of form factor tables
12
3-9
Blank
7X
10-18
a*
19-27
b*
28-36
c*
37-45
a*
46-54
p*
It
55-63
y*
t
(-,
a
not-)
F9.4
a
(in degrees)
It
Form factor tables (same format as ORXLS).
One
set of four cards for each different kind of atom.
1-9
f for
*
64-71
6.
Card 2
Card 1
Col.
=
1.15
1.55
0
1.20
Scale factors (1 per card).
0
Card 3
.75
7F9.2,
F8. 2
Card 4
.35
00
.80
.40
0
There must be NS cards.
Col.
1-9
F9.4
7
7.
?7
Overall temperature factor
Col.
1-9
8.
F9.4
Atom parameters (may be used in ORXLS)
(One card for each atom in asymmetric unit)
Col.
1-2
Form factor table to be used
12
3-7
Blank
5X
8-14
x
F7.4
15-21
y
t
22-28
z
t
29-35
B or P
36-42
p22
43-49
p3 3
50-56
p1 2
p1 3
57-63
64-72
9.
p2 3
'Multiplicity. "1 This is the quantity M7 i n Table 4.
12F6.4
Col.
10.
1-6
Atom I
7-12
Atom 2
67-72
Atom 12
Atom name
Col.
1-6
Any BCD characters desired to correspond
to atom 1, etc.
7-12
Any BCD characters desired to correspond
to atom 2, etc.
12A6
-I r7 Q
67-72
Any BCD characters desired to correspond
to atom 12, etc.
Use as many of cards 9 and 10 as are needed
to describe all atoms.
11.
Key card
Col.
1-3
NPAR:
4-6
Blank
7-9
NVAR:
Total number of parameters
13
3X
Number of parameters varied
13
NVAR must always be less than the number of nonzero weight, non-rejected reflections.
12.
Parameter selection card
7211
Col.
1-72
0 if the corresponding parameter is not to be
varied, 1 if it is to be varied.
The following
order is to be observed: s , s2'...
M, x,
y, z,
,Yyz
Bl, .. ., M , x
1VV
BT represents either B7 or
s ,B
V
,B
V
.
p1,
p12i' P13i' P23L. Use as many cards as
necessary to describe NPAR above.
its need
not be punched for mode 1 calculation.
13.
Reflection data (may be used in ORXLS)
Col.
1-3
h
13
4-9
blank
6X
10-12
k
13
13-18
blank
6X
19-21
13
- 11-
1 '70
22-27
blank
28-36
F
37-45
sigma (program computes
46-47
number of scale factor to be used for this
0
6X
or F 2
F9.2
0
VW=
It
-)
0-
reflection
12
48-54
blank
7X
55-63
extra input i
F9.2
64-70
extra input 2
F7. 2
71
blank
iX
72
blank
I1
Use one card for each reflection.
Note that
identifying letters in col. 72 sometimes
used in ORXLS will cause this program
to stop.
14.
End of reflection deck
Col.
15.
1-71
blank
71X
72
1
11
Run termination card
Col.
1-71
blank
71X
72
Mode 1: If 0, the program will stop.
11
If 1, the program will attempt to read a
new set of data.
Mode 2:
If 0, the program will stop.
If 1, a new cycle will be begun either by
returning to the main program from subroutine CHAIN or by calling in SFLSQ2 as
a chain link, depending on the way the run
is set up.
Submitting a run.
The following main program and subroutine
decks (left) should be included in the run deck (right).
SFSLQ3 (main program)
*
ID card
READ1
*
XEQ
READ2
SFLSQ3
READ3
Subroutines
LOOKUP
*
DATA
WEIGHT
Data deck
SPGRP
Data deck for additional cycles.
REJECT
STAT
PRSF
STMAT/SET
CHLNK3
SMI
TEST
CARD
DIST
MIFLIP
MIIOU
Tape B5 is always used by the program and may be saved as input to
further computations.
References
Busing, William R.,
and Henri A. Levy (1959),
A Crystallographic
Least-squares Program for the IBM 704. (Oak Ridge National
Laboratory, Central Files No. 59-4-37, Oak Ridge, Tenn.)
Busing, William R.,
and Henri A. Levy (1962), A Procedure for
Inverting Large Symmetric Matrices.
Comm. Assoc. Computing
Mach. In press.
Burnham, Charles W.,
and M. J. Buerger (1961),
crystal structure of andalusite.
Refinement of the
Zeit. Krist. 115, 269-290.
Cruickshank, D. W. J., Deana E. Pi.lling, A. Bujosa, F. M. Lovell,
and Mary R. Truter (1961),
In Computing Methods and the Phase
Problem in X-Ray Crystal Analysis, Pergamon Press, Oxford,
32-78.
41
Evans, Howard T.,
Jr. (1961),
Weighting factors for single-crystal
x-ray diffraction intensity data.
Geller, S. (1961),
refinement.
Acta Cryst. 14, 689.
Parameter interaction in least squares structure
Acta Cryst. 14, 1026-1035.
Hamilton, W. C. (1959),
On the isotropic temperature factor equiva-
lent to a given anisotropic temperature factor.
Acta Cryst. 12,
609-610.
Hybl, Albert, and Richard E. Marsh (1961),
Structure factor and
least-squares calculation for orthorhombic systems with
anisotropic vibrations.
Acta Cryst. 14, 1046-1051.
International Tables for X-Ray Crystallography.
(1959),
Vol. III (1962).
Levy, Henri A. (1956),
Vol. II
The Kynoch Press, Birmingham, England.
Symmetry relations among coefficients of the
anisotropic temperature factor.
U
Vol. 1 (1952),
Acta Cryst. 9, 679.
Rollett, J. S.,
and David R. Davies (1955),
The calculation of structure
factors for centrosymmetric monoclinic systems with anisotropic
atomic vibrations.
Rossmann, Michael G.,
Acta Cryst. 8, 125-128.
Robert A. Jacobson, F.
L. Hirshfeld, and
William N. Lipscomb (1959), An account of some computing
experiences.
Sly, W. G.,
D. P.
Acta Cryst. 12, 530-535.
Shoemaker, and J. H. Van den Hende (1962),
Two- and Three-dimensional Crystallographic Fourier
Summation Program for the IBM 7090 Computer. (Massachusetts
Institute of Technology, Esso Research and Engineering Company).
Trueblood, Kenneth N. (1956),
Symmetry transformations of general
anisotropic temperature factors.
Acta Cryst.,
9, 359-361.
183
L IST
LABEL
*
*
CSFLSQ3
C
MAIN PROGRAM FOR STRUCTURE FACTOR AND LEAST SQUARES REFINEMENT
C
THIS VERSION STORES THE NORMAL EQUATION MATRIX DIRECTLY IN CORE
C
6/7/62
COMMON CMACMBCMCCMDCMECMFAR,V
EQUIVALENCE (CMATITLE),(CMA(13),MODE) (CMA(14),INV),(CMA(15),ISAN
1),(CMA(16),NPU),(CMA(17),NEW),(CMA(22),NU),(CMA(23),NS),(CMA(24),A
2),(CMA(25)9B)9(CMA(26),C),(CMA(27),ALPHA),(CMA(28),BETA),(CMA(29),
3GAMMA),(CMA(30),NFOUR),(CMA(31),NCOR),(CMA(32),SUMDL),(CMA(33),KAR
4D),(CMA(34),ID),(CMA(35),NFSQ)
EQUIVALENCE (CMBNPAR),(CMB(2),NVAR),(CMB(3),NCOUNT),(CMB(4),KSEL)
EQUIVALENCE
(CMC'S),(CMC(51)BO)'(CMC(52)'MF),(CMC(102),X),(CMC
1(152),Y),(CMC(202),Z), (CMC(252),Bll-)BI),(CMC(302),B22),(CMC(352),
2B33),(CMC(402)9B12),(CMC(452),B23),(CMC(502),B13),(CMC(552),SYM),(
3CMC(602),NAME)
EQUIVALENCE (CMDNF),(CMD(2)'FORM),(CMD(642),F),(CMD(662),ASTER),(
1CMD(666),RHO),(CMD(667),RHOSQ),(CMD(668),ARG),(CMD(700),ISAVE)
EQUIVALENCE (CMEFOBS), (CME(2) ,FCAL) '(CME(3) ,AOBS) (CME(4) 'ACAL),(
1CME(5),BOBS),(CME(6),BCAL),(CME(7),DELTA),(CME(8),SIGMA),(CME(9),E
2XTi),(CME(10),EXT2),(CME(ll),SQRTW),(CME(12),MS1),(CME(13),MREJ),(
1
2
B3
B
B
B
4
3CME(14),MH),(CME(15),MK),(CME(16),ML),(CME(17),TH)'(CME(18),TK),(C
4ME(19),TL),(CME(20)'DA),(CME(220),DB)
EQUIVALENCE (CMF SFC) (CMF(2) SMSFC),(CMF(3),SQDL),(CMF(4),DELTAl)
1,(CMF(5),SUMF01),(CMF(6),WDL1),(CMF(7),WFo1),(CMF(8),DELTA2),(CMF(
29),SUMF02),(CMF(10),WDL2),(CMF(11),WF02)
DIMENSION CMA(50) CMB(507)CMC(651),CMD(700),CME(419),CMF(50)
DIMENSION TITLE(12),NEW(5)
DIMENSION FORM(32,20)tF(20),ASTER(4),ARG(32)
DIMENSION S(50),MF(50),X(50),Y(50),Z(50),BI(50),Bil(5O),B22(50),B3
13 (50) '812(50) 'B23 (50) ,B13(50) 9SYM(50) ,NAME(50)
DIMENSION DA(200),DB(200)pDF(200)
DIMENSION AR(14969) V(200)
DIMENSION KSEL(504)
CALL READ1
IF(NEW(2))2,3,2
CALL READ2
ASTER(1)=605454735460
ASTER(2)=605460606060
ASTER(3)=605454606060
ASTER(4)=606060606060
CALL READ3
ARG(32)=0.0
DO 4 I=1,31
J=32-I
K=33-I
ARG(J)=ARG(K)+.05
CONTINUE
ISAN=ISAN
MODE=MODE
INV=INV
ISAVE=1
NCOUNT=O
SUDL=0.0
184
400
401
402
403
404
405
406
407
500
501
6
8
9
10
11
12
13
14
15
16
180
181
182
183
21
22
23
DO 400 I=1,50
CMF(I)=0.0
GO TO (4079402),MODE
NT=NVAR+1
N=NVAR*NT
N=N/2
DO 404 I=1N
AR(I)=0.0
DO 406 1=1NVAR
V(I)=0.0
CALL SET(N*DFWDEL)
PI=3.1415927
RAD=PI/180.0
ALPHAl=ALPHA*RAD
BETA1=BETA*RAD
GAMMAl=GAMMA*RAD
AA=A*A/4.0
BB=B*B/4.0
CC=C*C/4.0
ABG=A*B*COSF(GAMMA1)/2.0
BCA=B*C*COSF(ALPHA1)/2.0
CAB=C*A*COSF(BETA1)/2.0
GO TO(6,69500),ISAN
DO 501 I=1,NU
B22(I)=BI(I)*BB
B33(I)=BI(I)*CC
B12(I)=BI(I)*ABG/2.0
B13(I)=BI(I)*CAB/2.0
B23(I)=BI(I)*BCA/2.0
B1l(I)=BI(I)*AA
CONTINUE
REWIND 7
IF(NEW(5))10,9,10
READ TAPE 10,MHMKMLFUBSSIGMAMSIEXT1,EXT2,LASTRHORHOSQF
GO TO 15
READ INPUT TAPE 4,11,MHMKML,F0LS,5IGMA,MSEXT1,EXT2,LAST
FORMAT(3(I3,6X),2F9.2,I2,7XF9.2,F7.2,1X,Il)
FREQUENCY 12(1,591)913(15.1),15(1,100,1)
IF(MS)13,15,13
IF(MS-MS1)14,15,14
MSl=MS
IF(LAST)5b,16,b
TH=MH
TK=MK
TL=ML
IF(NEW(5))182,181,182
IF(NEW(2))182,183,182
RHOSQ=TH*TH*AA+TK*TK*BB+L*TL*CC+TH*TK*ABG+TK*L*BCA+TL*TH* CAB
RHO=SQRTF(RHOSO)
CALL LOOKUP
WRITE TAPE 7,MHMKMLFOBSSIGMAMS1,EXT1,EXF2,LA$T ,RHORHO6QF
CALL WEIGHT
FREQUENCY 22(1,25,1)
IF(S16MA)23,24,23
SQ TW=1.0/SIGMA
185
24
25
26
27
28
29
30
31
34
340
341
342
35
36
37
38
39
40
41
42
43
44
450
451
46
47
48
GO TO 25
SQRTW=0.0
TO=EXPF(-BO*RHOSQ)
CALL SPGRP
MSl=MS1
GO TO(27,30),INV
Q=2.0*S(MS1)*TO
SFC=Q*ACAL
Q=SFC*SQRTW
IF(NFSQ)29,34,29
SFC=SFC**2
Q=2.0*SFC*SQRTW
GO TO 34
Q=S(MS1)*TO
FCSQ=ACAL**2+BCAL**2
FCAL=SQRTF(FCSQ)
SFC=Q*FCAL
Q=SFC*SQRTW
IF(NFSQ)29,34,29
CALL REJECT
FREQUENCY 340(1090,10)
IF(SFC)342,341,342
MREJ=0
CALL STAT
CALL PRSF
FREQUENCY 35(091925),36(0s1,20)
IF(MREJ)36,8,36
IF(SQRTW)37,8,37
NCOUNT=NCOUNT+1
SUMDL=SUMDL+SQDL
GO TO(8,38),MODE
K=0
DO 43 J=1,NS
IF(KSEL(J))40,43,40
K=K+1
IF(MS1-J)41,42,41
DF(K)=0.0
GO TO 43
DF(K)=Q/S(MS1)
CONTINUE
J=NS+1
IF(KSLL(J))44,450U44
K=K+1
DF(K)= -RHOSQ*Q
FREQUENCY 450(1,1950)
IF(NVAR-K)52,52,451
I=1
K=K+1
GO TO(47,49),INV
Cl=Q/ACAL
DO 48 J=KNVAR
DF(J)=DA(I)*C1
I=I+1
(CONTINUE
GOA TO 52
49
50
51
52
53
55
550
551
552
553
554
555
56
560
57
580
581
582
583
584
59
60
61
62
*
*
C
Cl=Q/FCSQ
DO 51 J=KtNVAR
DF(J)=(ACAL*DA(I)+BCAL*DB(I))*C1
I=I+1
CONTINUE
WDEL=SQRTW*DELTA
CALL STMAT
GO TO 8
WRITE TAPE 7,MHsMKMLFOBSSIGMAMSlEXT1EXT2,LAST RHO.RHOSQF
REWIND 7
REWIND 10
GO TO(552,56),MODE
WRITE TAPE 10NFABCALPHABETAGAMMASSEBOMFXYZBllB229B339
1812.B13,B23.SYM.NAMENPAR.NVAR.KSEL
READ TAPE 79MHMKMLFOBS SIGMAMS1,EXTiEXT2 LAST RHO.RHOSQF
WRITE TAPE 10MH.MK.MLFOBSSIGMAMS1,EXTlsEXT2,LASTgRH0sRHOSQ.F
IF(LAST)555*553,555
REWIND 7
REWIND 10
CALL PRST
IF(NPU)57,580,57
END FILE 3
IF(NFOUR)581,584,581
WRITE OUTPUT TAPE 9,582
FORMAT(1H171X)
IF(NFOUR-1)583,583,584
REWIND 9
GO TO(59961),MODE
READ INPUT TAPE 4,609IEND
FORMAT(71X9Il)
IF( IEND)1962ol
CALL CHAIN(2#B2)
GO TO 1
CALL EXIT
END
LIST
LABEL
SUbROUTINE READ1
READ INITIAL INPUT DATA AND WRITE HEADINGS ON OUTPUT TAPE
CUMMUN CMACMB,9(M.,(MD
EQUIVALENCE (CMATITLE),(CMA(13),MODE),(CMA(14),INV)9(CMA(15). ISAN
1),(CMA(16),NPU),(CMA(17),NEW),(CMA(22),NU),(CMA(23),N5).(CMA(24),A
2),(CMA(25),B),(CMA(26)tC).(CMA(27),ALPHA),(CMA(28),BETA),(CMA(29),
3GAMMA),(CMA(30),NFOUR),(CMA(31)NCOR),(CMA(32)6UMDL)(CMA(33)KAR
4D),(CMA(34)pID)v(CMA(35)qNFSQ)
T.Mb (4fltL)
EQUIVALENCL (CMbNPAR)9 (CML(2) NVAR) (CMB(3)9NGUUNI )91
(CMCS),(CMC(51)BO)(CMC(52),MF),(CMC(102),X),(CMC
EQUIVALENCE
1(152)9Y)#(CMC(202)q4)9 (CMC(252) Bll BI) (CMC(3U2)9b22)1(CMC(352)'
2B33),(CMC(402),B12).(CMC(452),B23),(CMC(502),B13),(CMC(552)9SYM),(
3CMC(602)#NAME)
EQUIVALENCE (CMDNF),(CMD(2),FORM),(CMD(642),F),(CMD(662),ASTER).(
1CMD(666),RHO),(CMD(667)HUQ),(CMD(668)9ARG)9(CMDV(700)9IAVt)
DIyENSION CMA(50),CMB(507),CMC(651),CMD(700)
187
DIMENSION TITLE(12),NEW(5)
DIMENSION S(50),MF(50),X(50),Y(50),Z(50),BI(50),B11(50)'B22(50)'83
13(50),B12(50),B23(50),B13(50),SYM(50),NAME(50)
DIMENSION KSEL(504)
DIMENSION FLP(4)
READ INPUT TAPE 4,1,TITLE
1
FORMAT(12A6)
READ INPUT TAPE 4,2,FLP(3),FLP(4),FLP(1),FLP(2)
2
FORMAT(4A6)
WRITE OUTPUT TAPE 2,3,TITLE
3
FORMAT(lHll2A6)
READ INPUT TAPE 4,49MODEINVISANNPUNEWNUNSNFOURNCORKARDID
1,NFSQ
4
FORMAT(9Il,2I2,5II)
WRITE OUTPUT TAPE 295
5
FORMAT(3OHOSENSE CARD AS READ BY PROGRAM)
WRITE OUTPUT TAPE 2,6,MODEINVISAN.NPUNEWNUNSNFOURNCORKARD,
1IDtNFSQ
6
FORMAT(lHO9Il92I2,5Il)
WRITE OUTPUT TAPE 2,3PTITLE
REWIND 10
7
IF(NEW(1))8,12,8
IF(NEW(2))9,12,9
8
9
IF(NEW(3))10,12,10
10
IF(NEW(4))11,12J11
11
IF(NEW(5))14,12,14
12
READ TAPE 10BNF#A#BCALPHA,8ETAGAMMA*SB0,MFXYZBllB22,B33,B
ll2,B13,B23,SYMNAMENPARNVARKSEL
13
IF(NEW(1))14,16,14
14
READ INPUT TAPE 4,15,NF,A,B,C,ALPHABLTAGAMMA
15
FORMAT(I2#7X,6F9,4)
16
IF(NPU+KARD)17,18,17
CALL PILF1(FLP,4)
17
18
IF(NFSQ)181,180,181
8180
OBS=602646226260
B
CAL=602623214360
GO TO 182
B181 OBS=462262545402
B
CAL=232143545402
182 GO TO (19,21),INV
19
WRITE OUTPUT TAPE 2,20,OBSCAL
20
FORMAT(18HO H
K
L
A6,/H
A6,5/H
AUB5
1 ACAL
OBS-CAL
(OBS-CAL)/SIGMA)
GO TO 23
21
WRITE OUTPUT TAPE 2922,OBSCAL
22
FORMAT(18HO
H
K
L
A6,7H
A6983H
AUBS
1 ACAL
BOBS
BCAL
OBS-CAL
(OBS-CAL)/SIGMA)
23
IF(NFOUR)24,25,24
WRITE OUTPUT TAPE 9,1,TITLE
24
RETURN
25
END
*
LIT
LAPEL
188
CREAD2
C
5
6
SUBROUTINE READ2
READ FORM FACTOR FACTOR TABLES
COMMON CMA#CMB,CMCCMD
EQUIVALENCE (CMATITLE) (CMA(13)tMODE)t(CMA(14),INV),(CMA(15)s ISAN
2)(CMA(25)tB)t(CMA(26)C)9(CMA(27),ALPHA),(CMA(28),BETA)(CMA(29),
3GAMMA),(CMA(30),NFOUR),(CMA(31),NCOR),(CMA(32),SUMDL)
EQUIVALENCE (CMBtNPAR)
EQUIVALENCE (CMC#S)
EQUIVALENCE (CMDNF).(CMD(2),FORM)9(CMD(642),F),(CMD(662),ASTER),t(
1CMD(666)tRHO),(CMD(667),RHOSQ),(CMD(668),ARG),(CMD(700),ISAVE)
DIMENSION CMA(50),CMB(507),CMC(651),CMD(700)
DIMENSION FORM(32,20),F(20),ASTER(4)tARG(32)
DO 6 J=lNF
READ INPUT TAPE 4,5q(FORM(I#J),I=1,32)
FORMAT(7F9.2,F8.2)
CONTINUE
RETURN
END
LIST
LABEL
*
*
C
1
2
3
4
5
SUBROUTINE READ3
SUBROUTINE FOR READING ATOM PARAMETERS
COMMON CMACMBCMC
EQUIVALENCE (CMATITLE),(CMA(13),MODE)t(CMA(14),INV),(CMA(15),ISAN
1),(CMA(16)tNPU),(CMA(17),NEW),(CMA(22),NU),(CMA(23),NS).(CMA(24),A
2)*(CMA(25)B) (CMA(26) C) (CMA(27)tALPHA) (CMA(28) BETA) (CMA(29),
3GAMMA),(CMA(30),NFOUR),(CMA(31),NCOR),(CMA(32),SUMDL),(CMA(33),KAR
4D)#(CMA(34)#ID)
EQUIVALENCE (CMBtNPAR) (CMB(2) NVAR) (CMB(3) NCOUNT),(CMB(4),KSEL)
(CMCS),(CMC(51),BO),(CMC(52),MF),(CMC(102),X),(CMC
EQUIVALENCE
1(152)tY)#(CMC(202)tZ), (CMC(252) Bl1BI ) (CMC(302)tB22)t(CMC(352)'
2B33),(CMC(402),B12),(CMC(452),B23),(CMC(502)9B13),(CMC(552)'SYM)(
3CMC(602)#NAME)
DIMENSION TITLE(12),NEW(5)
DIMENSION $(50),MF(50),X(50)tY(50)9,(50),BI(50)9811(50)9B22(50)983
13(50) B12(50)9B23(50) B13(50),SYM(50),NAME(50)
DIMENSION KSEL(504)
DIMENSION CMA(50),CMB(507),CMC(651)
IF(NEW(3))2.9,2
DO 4 I=lNS
READ INPUT TAPE 4,3*S(I)
FORMAT(F9.4)
CONTINUE
READ INPUT TAPE 4t5,BO
FORMAT(F9.4)
READ INPUT TAPE 4,6,(MF(I),X(I) Y(I),Z(I),Bll(I),B22(I),B33(I),B12
1(I)ipB13(I)tB23(I),I=leNU)
6
7
8
FORMAT(I2,5X,9F7.4)
READ INPUT TAPE 4,7,(SYM(I)tI=1=NU)
FORMAT(12F6.4)
READ INPUT TAPE 4,8v(NAME(I)*I=l#NU)
FOFMAT(12A6)
189
9
10
11
12
13
14
*
*
LIST
LABEL
C
1
2
3
4
5
6
8
10
11
12
13
20
21
22
23
24
30
31
C
32
33
34
C
IF(NEW(4))10,14,10
READ INPUT TAPE 4,11.NPARtNVAR
FORMAT(I3,3XI3)
READ INPUT TAPE 4,13,(KSEL(J),J=1,NPAR)
FORMAT (7211)
RETURN
END
SUBROUTINE LOOKUP
TABLE LOOK-UP FOR FORM FACTORS
COMMON CMAoCMB.CMCPCMD
EQUIVALENCE (CMAtTITLE)
EQUIVALENCE (CMBtNPAR)
EQUIVALENCE (CMCS)
EQUIVALENCE (CMDNF),(CMD(2),FORM),(CMD(642),F),(CMD(662),ASTER),(
1CMD(666),RHO),(CMD(667),RHOSQ),(CMD(668),ARG),(CMD(700),ISAVE)
DIMENSION FORM(32,20),F(20),ASTER(4),ARG(32)
DIMENSION CMA(50),CMB(507),CMC(651),CMD(700)
FREQUENCY 1(OO1)'2(0,0,1),4(1,O,1),6(1,O,10),8(5),12(5),21(10,O0
11)22(5)t30(5)p37(5)
IF(RHO)32,30,2
IF(1.55-RHO)35,37,3
I=ISAVE
IF(RHO-ARG(I))20,12,5
I=1-i
IF(RHO-ARG(I))8,12,5
DO 10 J=1,NF
F(J)=(FORM(I+1,J)-FORM(IJ))*(ARG(I)-RHO)/.05+FORM(I.J)
CONTINUE
ISAVE=I
RETURN
DO 13 J=1,NF
F(J)=FORM(ItJ)
CONTINUE
ISAVE=I
RETURN
1=1+1
IF(RHO-ARG(I))20,12,22
DO 23 J=1,NF
F(J)=(FORM(IJ)-FORM(I-1,J))*(ARG(I-1)-RHO)/.05+FORM(I-1,J)
CONTINUE
ISAVE=I
RETURN
DO 31 J=1,NF
F(J)=UKM(32 ,J)
CONTINUE
RETURN
NEGATIVE RHO
WRITE OUTPUT TAPL 2933
FORMAT(3OHONEGATIVE RHO - RUN TERMINATED)
CALL ExIT
RH0 GREATER THAN 1.55
190
35
36
37
38
WRITE OUTPUT TAPE 2.36
FORMAT(39HORHO GREATER THAN 1.55
CALL EXIT
DO 38 J=1NF
F(J)=FORM(1,J)
CONTINUE
RETURN
END
-
RUN TERMINATED)
LIST
LABEL
SYMBOL TABLE
*
*
*
C
100
200
201
202
203
SUBROUTINE SPGRP
GENERAL SPACE GROUP ROUTINE
COMMON CMAtCMBCMCPCMDoCME
EQUIVALENCE (CMA.TITLE),(CMA(13),MODE),(CMA(14),INV)#(CMA(15),ISAN
1),(CMA(16),NPU).(CMA(17) NEW),(CMA(22),NU) (CMA(23),NS),(CMA(24),A
2)t(CMA(25),B),{CMA(26),C),(CMA(27),ALPHA),(CMA(28)9BETA)9(CMA(29)9
3GAMMA),(CMA(30),NFOUR),(CMA(31),NCOR),(CMA(32),SUMDL),(CMA(33),KAR
4D),(CMA(34),ID)
EQUIVALENCE (CMBNPAR).(CMB(2) NVAR),(CMB(3),NCOUNT),(CMB(4),KSEL)
(CMC.S),(CMC(51),BO),(CMC(52),MF).(CMC(102) X),(CMC
EQUIVALENCE
1(152)#Y)#(CMC(202),Z), (CMC(252)B.11,BI),(CMC(302) B22),(CMC(352)'
2B33),(CMC(402),B12)*(CMC(452),B23),(CMC(502)9B13),(CMC(552)9SYM)*(
3CMC(602),NAME)
EQUIVALENCE (CMDNF),(CMD(2),FORM),(CMD(642)9F),(CMD(662),ASTER),(
1CMD(666),RHO),(CMD(667),RHOSQ),(CMD(668),ARG),(CMD(700),ISAVE)
EQUIVALENCE (CMEFOBS),(CME(2),FCAL),(CME(3)9AOBS),(CME(4)9ACAL),(
1CME(5),BOBS),(CME(6),BCAL)(CME(7)DELTA),(CME(8),SIGMA)9(CME(9),E
2XT1),(CME(10),EXT2),(CME(11),SQRTW)9(CME(12)9MS1),(CME(13),MREJ),(
3CME(14),MH),(CME(15),MK),(CME(16),ML),(CME(17),TH)9(CME(18),TK),(C
4ME(19) TL) (CME(20),DA),(CME(220),DB)
DIMENSION TITLE(12),NEW(5)
DIMENSION FORM(32,20),F(20),ASTER(4),ARG(32)
DIMENSION S(50)#MF(50),X(50),Y(50)#Z(50)tBI(50),B11(50)9B22(50),B3
13(50) B12(50) B23(50) B13(50) SYM(50),NAME(50)
DIMENSION KSEL(504)
DIMENSION DA(200),DB(200)
DIMENSION CMA(50),CMB(507),CMC(651)9CMD(700)9CME(419)
INV=INV
ISANISAN
MODE=MODE
ACAL=0.0
BCAL=O.0
K=1
L=NS+2
TPI=6*2831854
DO 700 I=1NU
JJ=MF(I)
SMCP=0.0
SMSP=o.0
GO TO(300,203),MODE
CPH=O0
CP6=0.0
191
204
205
C
300
C
C
301
302
303
304
305
306
307
400
401
402
403
404
405
CPL=O.0
SPH=0.0
SPK=0.0
SPL=0.0
GO TO(300#205,205),ISAN
CPHH=0.0
CPKK=0.0
CPLL=0.0
CPHK=0.0
CPHL=0.0
CPKL=O.O
SPHH=0.0
SPKK=0.0
SPLL=O.0
SPHK=0.0
SPHL=O.0
SPKL=0.0
LOOP FOR EACH ATOM IN ASYMMETRIC UNIT
DO 414 J=1,6
INDEX TRANSFORMATION SECTION
SPACE GROUP R 3BAR C
GO TO(302,303,304,305,3069307),J
RH=TH
RK=TK
RL=TL
T=0.0
GO TO 400
RH=TK
RK=-(TH+TK)
GO TO 400
RH=-(TH+TK)
RK=TH
GO TO 400
RH=TK
RK=TH
RL=-TL
T=TL/2.0
GO TO 400
RH=-(TH+TK)
RK=TK
GO TO 400
RH=TH
RK=-(TH+TK)
PHI=TPI*(RH*X(I)+RK*Y(I)+RL*Z(I)+T)
GO TO(402#407.9407),ISAN
CP=COSF(PHI)
SP=SINF(PHI)
EQUIVALENCE (CPHSPH),(CPK,SPK),(CPLSPL),(SMCPsMSP)
EQUIVALENCE(CPHHSPHH),(SPKKSPKK),(CPLLSPLL),(CPHKSPHK)
EQUIVALENCE(CPHL.SPHL),(CPKLSPKL)
GO TO (413,404),MODE
SPH=SPH+SP*RH
SPK=SPK+SP*RK
SPL=SPL+SP*RL
GOTO (413,406),INV
192
406
CPH=CPH+CP*RH
CPK=CPK+CP*RK
CPL=CPL+CP*RL
GO TO 413
407 GO TO(407094071),I
4070 RHH=RH**2
RKK=RK**2+RH*RK
RLL=RL**2
RHK=0.0
RHL=0.0
RKL=2*0*RK*RL+RH*RL
GO TO 408
4071 RHH=RH**2+RK**2+RH*RK
RKK=0.0
RLL=RL**2
RHK=0.0
RHL=0.O
RKL=0.0
408 E=EXPF(-(RHH*B11(I)+RKK*B22(I)+RLL*B33(I)+RHK*B12(I)+RHL*B13(I)+RK
1L*B23(I)))
CP=COSF(PHI)*E
SP=SINF(PHI)*E
409 GO TO(413,410),MODE
410 SPH=SPH+SP*RH
SPK=SPK+SP*RK
SPL=SPL+SP*RL
CPHH=CPHH+CP*RHH
CPKK=CPKK+CP*RKK
CPLL=CPLL+CP*RLL
CPHK=CPHK+CP*RHK
CPHL=CPHL+CP*RHL
CPKL=CPKL+CP*RKL
411 GO TO (413#412)tINV
412 CPH=CPH+CP*RH
CPK=CPK+CP*RK
CPL=CPL+CP*RL
SPHH=SPHH+SP*RHH
SPKK=SPKK+SP*RKK
SPLL=SPLL+SP*RLL
SPHK=SPHK+SP*RHK
SPHL=SPHL+SP*RHL
SPKL=bPKL+5P*RKL
413
SMCP=SMCP+CP
SMSP=SMSP+SP
414 CONTINUE
500 GO TO(50195029502)#ISAN
501
TMP=EXPF(-BI(I)*RHOSQ)*SYM(I)
GO TO 503
502
TMP=SYM(I)
503 GO TO(5049505)oI
504 TMPA=(F(JJ)-ll)*TMP
TMPB=3.4*TMP
GO TO 506
TMPA=F(JJ)*TMP
505
TMFB=0.0
193
506
507
508
600
601
602
603
604
605
606
607
608
609
610
AR=TMPA*SMCP
AI=TMPB*SMCP
ACAL=ACAL+AR
BCAL=BCAL+AI
GO TO(700,600),MODE
TA=TPI*TMPA
TB=TPI*TMPB
RA=-RHOSQ*TMPA
RB=-RHOSQ*TMPB
IF(NVAR-K)700,601,601
IF(KSEL(L))602,606,602
DA(K)=AR/SYM(I)
GO TO(605,604),INV
DB(K)=BR/SYM(I)
K=K+1
L=L+1
IF(KSEL(L))607,610,607
DA(K)=-TA*SPH
GO TO(609,608),INV
DB(K)=TB*CPH
K=K+1
L=L+1
IF(KSEL(L))611,615,611
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
DA(K)=-TA*SPK
GO TO(614,613),INV
D8(K)=TB*CPK
K=K+1
L=L+1
IF(KSEL(L))616,620#616
DA(K)=-TA*SPL
GO TO(619,618),INV
DB(K)=TB*CPL
K=K+1
L=L+1
GO TO(622,628,628)oISAN
IF(KSEL(L))623,627,623
DA(K)=RA*SMCP
GO TO(626,625),INV
DB(K)=RB*SMSP
K=K+1
L=L+1
GO TO 700
IF(KSEL(L))629,633,629
DA(K)=-TMPA*CPHH
GO TO(632,631),INV
D8(K)=-TMPB*SPHH
K=K+1
L=L+1
IF(KSEL(L))634,638,634
DA(K)=-TMPA*CPKK
GO TO(637,636),INV
DB(K)=-IMPB*SPKK
K=K+1
L=L+l
IF,(KSEL(L) )639,643,639
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
700
*
*
C
DA(K)=-TMPA*CPLL
GO TO(642,641),INV
DB(K)=-TMPB*SPLL
K=K+1
L=L+1
IF(KSEL (L) )644,648,644
DA(K)=-TMPA*CPHK
GO TO(647,646)#INV
DB(K)=-TMPB*SPHK
K=K+1
L=L+1
IF(KSEL(L))649,653,649
DA(K)=-TMPA*CPHL
GO TO(652o651)*INV
DB(K)=-TMPB*SPHL
K=K+1
L=L+1
IF(KSEL(L))654,658,654
DA(K)=-TMPA*CPKL
GO TO(657,656),INV
DB(K)=-TMPB*SPKL
K=K+1
L=L+1
CONTINUE
RETURN
END
LIST
LABEL
SUBROUTINE SPGRP
SPACE GROUP Pl BAR WITH DISPERSION CORRECTION FOR 2 CA.
COMMON CMA#CMBCMCCMDCME
EQUIVALENCE (CMA.TITLE),(CMA(13),MODE) (CMA(14)'INV),(CMA(15) ISAN
1),(CMA(16),NPU),(CMA(17),NEW),(CMA(22),NU),(CMA(23.),NS),(CMA(24),A
2),(CMA(25),B),(CMA(26),C),(CMA(27),ALPHA),(CMA(28),BETA),(CMA(29),
3GAMMA),(CMA(30),NFOUR),(CMA(31)9NCOR),(CMA(32),SUMDL),(CMA(33),KAR
4D)9(CMA(34),ID)
EQUIVALENCE (CMBNPAR),(CMB(2),NVAR),(CMB(3),NCOUNT),(CMB(4),KSEL)
(CMC,5),(CMC(51),BO),(CMC(52)OMF),(CMCIlU2),X)9(CMC
EQUIVALENCE
1(152)#Y),(CMC(202),Z), (CMC(252),B11BI).(CMC(302),B22),(CMC(352),
2B33),(CMC(402),B12),(CMC(452),B23)#(CMC(502),013),(CMC(552),5YM),(
3CMC(602)tNAME)
EQUIVALENCE (CMD'NF),(CMD(2),FORM),(CMD(642),F),(CMD(662),AbJER),(
1CMD(666),RHO),(CMD(667),RHOSQ),(CMD(668),ARG) (CMD(700),ISAVE)
EQUIVALENCE (CMEFOBS),(CME(2),FCAL),(CME(3),AOBS),(CME(4),ACAL),(
1CME(5),BOBS),(CME(6),BCAL),(CME(7),DELTA),(CME(8),SIGMA).(CME(9),E
2XT1),(CME(10),EXT2),(CME(11),SQRTW),(CME(12),M51),(CME(13)9MREJ),(
3CME(14),MH),(CME(15),MK),(CME(16),ML),(CME(17),TH),(CME(18)9TK),(C
4ME(19),TL),(CME(20)tDA),(CME(220) DB)
DIMENSION TITLE(12),NEW(5)
DIMENSION FORM(32,20),F(20),A5TER(4),ARG(32)
DIMENSION S(50) MF(50),X(50),Y(50),Z(50),BI(50)'811(50),B22(50),B3
13(50),B12(50),B23(50),B13(50),SYM(5U),NAME(50)
DIOENSION KSEL(504)
195
1
DIMENSION DA(200),DB(200)
DIMENSION CMA(50),CMB(507),CMC(651),CMD(700),CME(419)
ACAL=0.0
BCAL=0.0
IP=l
ISAN=ISAN
MODE=MODE
L=NS+2
K=1
C2=6*2831854
DO 27 I=1,NU
J=MF(I)
GO TO(101,102*102)tISAN
TMP=EXPF(-BI(I)*RHOSQ)*SYM(I)*2.0
100
101
1010 IF(IP-2)1011,1011,1012
1011 Cl=(F(J)+0.2)*TMP
D1=1.4*TMP
GO TO 103
1012 C1=F(J)*TMP
D1=0.0
GO TO 103
THS=TH*TH
102
TKS=TK*TK
TLS=TL*TL
THKS=TH*TK
TKLS=TK*TL
THLS=TH*TL
TMP=EXPF(-(THS*Bll(I)+TKS*822(I)+TLS*B33(I)+2.0*(THKS*B12(I)+THLS*
1B13(I)+TKLS*B23(I))))*SYM(I)*2.0
GO TO 1010
103 C3=C2*(TH*X(I)+TK*Y(I)+TL*Z(I))
C4=COSF (C3)
C5=SINF(C3)
AR=Cl*C4
AJ=D1*C4
ACAL=ACAL+AR
BCAL=BCAL+AJ
GO TO(26,200),MODE
200
IF(NVAR-K)26,201,201
IF(KSEL(L))202,203,202
201
DA(K)=AR/SYM(I)
202
DB(K)=AJ/SYM(I)
K=K+1
203
L=L+1
IF(KSEL(L))3,4,3
DT=-C2*TH*C5
3
DA(K)=DT*Cl
DB(K)=DT*D1
K=K+1
4
L=L+1
IF(KSEL(L) )5,6,5
DT=-C2*TK*C5
5
DA(K)=DT*Cl
DB( K)=DT*D1
K=6+1
196
6
7
8
9
10
11
12
13
14
15
L=L+1
IF(KSEL(L))7,8,7
DT=-C2*TL*C5
DA(K)=DT*C1
DB(K)=DT*D1
K=K+1
L=L+1
GO TO(10.13913)tISAN
IF(KSEL(L))11,12.11
DT=-RHOSQ*C4
DA( K) =DT*C1
DB( K)=DT*D1
K=K+1
L=L+1
GO TO 26
IF(KSEL(L))14915.14
DT=-THS*C4
DA(K)=Cl*DT
DB(K)=D1*DT
K=K+1
L=L+1
IF(KSEL(L))16.17o16
16
17
DT=-TKS*C4
DA(K)=Cl*DT
DB(K)=D1*DT
K=K+1
L=L+1
IF(KSEL(L))18919918
18
19
20
21
22
23
24
25
26
27
DT=-TLS*C4
DA(K)=Cl*DT
DB( K)=D1*DT
K=K+1
L=L+1
IF(KSEL(L))20921#20
DT=-THKS*C4
DA (K) =Cl*DT
DB(K)=D1*DT
K=K+1
L=L+1
IF(KSEL(L))22,23.22
DT=-THLS*C4
DA(K)=Cl*DI
DB( K) =D1*DT
K=K+1
L=L+1
IF(KSEL(L) )24,25924
DT=-TKLS*C4
DA( K)=Cl*DT
DB(K)=D1*DT
K=K+1
L=L+1
IP=IP+1
CONTINUE
RETURN
ENP
197
LIST
LABEL
*
*
C
C
1
2
5
6
*
*
SUBROUTINE WEIGHT
STORE STANDARD ULVIATION IN LOCATION SIGMA IF DIFFERENT THAN ON
DATA CARD. MAIN PROGRAM COMPUTES SQRT WEIGHT=l/SIGMA.
COMMON CMA#CMBCMC#CMDtCME
EQUIVALENCE (CMATITLE)
EQUIVALENCE (CMBPNPAR)
EQUIVALENCE (CMCS)
EQUIVALENCE (CMDNF),(CMD(2),FORM),(CMD(642),F),(CMD(662),ASTER),(
1CMD(666),RHO),(CMD(667),RHOSQ),(CMD(668),ARG),(CMD(700),ISAVE)
EQUIVALENCE (CMEFOBS),(CME(2),FCAL),(CME(3),AOBS),(CME(4),ACAL),(
1CME(5),BOBS),(CME(6),BCAL),(CME(7).DELTA),(CME(8),SIGMA),(CME(9),E
2XTl),(CME(10),EXT2),(CME(11),SQRTW),(CME(12),MS1),(CME(13),MREJ),(
3CME(14),MH),(CME(15),MK),(CME(16),ML),(CME(17),TH),(CME(18),TK),(C
4ME(19),lL),(CME(20),DA),(CME(220),LDb)
DIMENSION CMA(50),CMB(507),CMC(651),CMD(700),CME(419)
FREQUENCY 1(01,10)p2(2,0,1)
IF(FOBS)2,5,2
SIGMA=SQRTF(3.02+FOBS+.0193*FOBS**2)
GO TO 6
SIGMA=0.0
RETURN
END
LIST
LABEL
SUBROUTINE REJECT
REJECTION TEST FOR REFLECTION DATA. REJECT IF MREJ=O.
COMMON CMACMBCMCCMDCMECMF
EQUIVALENCE (CMATITLE)
EQUIVALENCE (CMBqNPAR)
EQUIVALENCE (CMCS)
EQUIVALENCE (CMDoNF)
EQUIVALENCE (CMEFOBS),(CME(2),FCAL),(CME(3),AOBS),(CME(4),ACAL),(
1CME(5),BOBS),(CME(6),BCAL),(CME(7),DELTA),(CME(8),SIGMA),(CME(9),E
2XTl),(CME(10),EXT2),(CME(11),SQRTW),(CME(12)9MS1)9(CME(13),MREJ),(
C
3CME(14),MH),(CME(15),MK),(CME(16),ML),(CME(17),TH),(CME(18),TK),(C
10
1
2
3
4
5
4ME(19),TL),(CME(20),DA),(CME(220),Db)
EQUIVALENCE (CMFSFC),(CMF(2),SMSFC),(CMF(3),SQDL),(CMF(4),DELTAl)
1,(CMF(5),SUMFO1),(CMF(6),WDL1),(CMF(7),WFO1),(CMF(8),DELTA2),(CMF(
29),SUMF02),(CMF(10),WDL2),(CMF(11),WF02)
DIMENSION CMA(50),CMB(507),CMC(651),CMD(700),CME(419),CMF(50)
IF(FOBS)1,4,1
IF(.333*FOBS-ABSF(SFC))2,4,4
IF(((ABSF(FOBS-ABSF(SFC)))/FOBS)-.20)3,4,4
MREJ=1
GO TO 5
MREJ=O
RETURN
END
198
*
*
LIST
LABEL
SUBROUTINE STAT
ACCUMULATION OF STATISTICAL DATA
COMMON CMAtCMBCMCCMDCMEgCMF
EQUIVALENCE (CMATITLE),(CMA(13),MODE),(CMA(14),INV),(CMA(15),ISAN
1),(CMA(16),NPU),(CMA(17),NEW),(CMA(22),NU),(CMA(23),NS),(CMA(24),A
2),(CMA(25),B),(CMA(26),C),(CMA(27),ALPHA),(CMA(28),BETA),(CMA(29),
3GAMMA),(CMA(30),NFOUR),(CMA(31),NCOR),(CMA(32),SUMDL)
EQUIVALENCE (CMBNPAR)
EQUIVALENCE (CMCS)
EQUIVALENCE (CMDNF)
EQUIVALENCE (CMEFOBS),(CME(2),FCAL),(CME(3),AOBS),(CME(4),ACAL),(
1CME(5),BOBS),(CME(6),BCAL),(CME(7),DELTA),(CME(8),SIGMA),(CME(9),E
2XT1),(CME(10),EXT2),(CME(11),SQRTW),(CME(12),MS1),(CME(13),MREJ),(
C
3CME(14),MH),(CME-(15),MK),(CME(16),ML),(CMIE(17),TH),(CME(18),TK),(C
1
10
11
12
13
3
4
5
*
*
C
4ME(19),TL),(CME(20),DA) (CME(220),DB)
EQUIVALENCE (CMFSFC)(CMF(2),SMSFC),(CMF(3),SQDL),(CMF(4),DELTA1)
1,(CMF(5),SUMFO1),(CMF(6),WDL1),(CMF(7),WFO1),(CMF(8),DELTA2),(CMF(
29),SUMF02),(CMF(10),WDL2),(CMF(11)#WF02)
DIMENSION CMA(50),CMB(507),CMC(651),CMD(700),CME(419),CMF(50)
SMSFC=SMSFC+ABSF(SFC)
INV=INV
GO TO(1112),INV
DELTA=SIGNF(FOBSACAL)-SIGNF(SFCACAL)
GO TO 13
DELTA=FOBS-SFC
SQDL=(SQRTW*DELTA)**2
SQFO=(SQRTW*FOBS)**2
DELTA1=DELTA1+ABSF(DELTA)
SUMFQ1=SUMFO1+ABSF(FOBS)
WDLl=WDL1+SQDL
WFO1=WFO1+SQFO
IF(FOBS)4,5,4
DELTA2=DELTA2+ABSF(DELTA)
SUMF02=SUMF02+ABSF(FOBS)
WDL2=WDL2+SQDL
WF02=WF02+SQFO
RETURN
END
LIST
LABEL
SUBROUTINE PRSF
SUBROUTINE FOR PRINTING STRUCTURE FACTORS
COMMON CMACMBCMC*CMDgCMEsCMF
EQUI.VALENCE (CMATITLE),(CMA(13),MODE),(CMA(14),INV),(CMA(15),ISAN
1),(CMA(16),NPU),(CMA(17),NEW),(CMA(22),NU),(CMA(23)9NS)9(CMA(24),A
2),(CMA(25),B),(CMA(26),C),(CMA(27),ALPHA),(CMA(28),BETA),(CMA(29),
3GAMMA),(CMA(30),NFOUR),(CMA(31),NCOR),(CMA(32),SUMDL)
EQUIVALENCE (CMBNPAR)
EQPIVALENCE (CMCS)
199
EQUIVALENCE (CMDNF),(CMD(2),FORM),(CMD(642),F),(CMD(662),ASTER),(
1CMD(666),RHO),(CMD(667),RHOSQ),(CMD(668),ARG),(CMD(700),ISAVE)
EQUIVALENCE (CMEFOBS),(CME(2),FCAL),(CME(3)'AOBS),(CME(4),ACAL),(
ICME(5),BOBS),(CME(6),BCAL),(CME(7),DELTA),(CME(8),SIGMA),(CME(9),E
2XT1),(CME(10),EXT2),(CME(11),SQRTW),(CME(12),MS1),(CME(13),MREJ),(
3CME(14),MH),(CME(15),MK),(CME(16),ML),(CME(17),TH),(CME(18),TK)9(C
4ME(19),TL),(CME(20),DA),(CME(220),DB)
EQUIVALENCE (CMFSFC),(CMF(2),SMSFC),(CMF(3),SQDL),(CMF(4),DELTA1)
1,(CMF(5),SUMFO1),(CMF(6),WDL1),(CMF(7),WFO1),(CMF(8),DELTA2),(CMF(
1
18
19
29),SUMF02),(CMF(10),WDL2),(CMF(11),WF02)
DIMENSION TITLE(12),NEW(5)
DIMENSION CMA(50),CMB(507),CMC(651),CMD(700),CME(419),CMF(50)
DIMENSION FORM(32,20),F(20),ASTER(4),ARG(32)
DIMENSION S(50)
FREQUENCY 21(10,1,10),22(1,1,5)925(1,1,5)
INV=INV
MSl=MS1
GO TO(18,19),INV
AOBS=FOBS/S(MS1)
FOSC=AOBS
AOBS=SIGNF(AOBSACAL)
GO TO 20
C1=FOBS/SFC
FOSC=FOBS/S(MS1)
A0BS=Cl*ACAL
20
21
22
25
31
32
33
34
5
51
6
7
8
9
10
11
12
13
61
62
63
64
65
BOBS=Cl*BCAL
DS=DELTA/SIGMA
IF DIVIDE CHECK 21,21
IF(MREJ)25,22,25
IF(2.0-ABSF(DS))31,31,32
IF(2.0-ABSF(DS))33,33,34
I=1
GO TO 5
I=2
GO TO 5
1=3
GO TO 5
I=4
GO TO(51,61),INV
WRITE OUTPUT TAPE 2,6,MHMKMLFOB5,5FCAOBSACALDELTAD5,ASTER(I
1)
FORMAT(3I4,93X,4(F9.2,4X),F9.2,3X.F9.2.A6)
IF(NPU)8,10,8
PUNCH 9,MHMKMLAOBSACALFOSCASTER(I)
FORMAT(3I4,12X,2F8.2,16XF8.2,8XA6)
IF(NFOUR)11,13911
WRITE OUTPUT TAPE 9,12,MHMKMLAOBSACALFOSC
FORMAT(314,12X,268.2,16Xd8.2,A)
RETURN
WRITE OUIPUT TAPE 2,62,MHgMKML#FOB5,5FCAOB5,ACALBOBbBCALDELIA
1,DStASTER(I)
FORMAl(314,3X,6( 9.2,4X),F9@2,3A,69.2,A6)
IF(NPU)64,66,64
bAltKil)
PUNCH 65,MHMKMLAUB5,ACALb,00bBCALgb
FORMAT(3I4,12X,5F8.2,8XA6)
200
66
67
68
69
*
IF(NFOUR)67,69,67
WRITE OUTPUT TAPE 9,68,MHMKMLAOBSACALBOBS.BCALFOSC
FORMAT (314,12X95F8.2,8X)
GO TO 13
END
FAP
COUNT
REM
ENTRY
ENTRY
CLA
SET
STA
CLA
ADD
STA
STA
STA
CLA
STA
CLA
STD
STD
TRA
STMAT SXD
$XD
SXD
CLA
TA
SUB
PDX
LXD
PXD
AA
PDX
REM
LDQ
BA
FMP
FAD
STO
TXI
TXI
IXL
EB8
REM
LDQ
FMP
FAD
STO
TXI
TXL
BC
LXD
LXD
LXD
TRA
PZE
ONE
BgS
SA
50
ROUTINE FOR STORING NORMAL EQUATION MATRIX
SET
STMAT
STORE ADDRESSES FOR ARGUMENTS
194
TA
2,4
=1
BA
BA+1
BB+1
3,4
BB+2
NVAR
BB
BC
4,4
SA,1
SA+1,2
SA+2,4
N
ONE
0,4
ONE,1
0,1
0,2
LOOP TO FORM AND STORE MATRIX ELEMENTS
DF+1
0,1
DF+1
0,2
AR,4
AR,4
*+1,2,1
*+1#49-1
BA92
LOOP TO FORM AND STORE VECTOR ELEMENTS
DF+1
0,1
WDEL
V+1,1
V+1,1
*+1,1,1
AA91
SA,1
SA+1,2
SA+2,4
1,4
0,0,1
3
201
NVAR
AR
V
*
*
COMMON
COMMON
COMMON
COMMON
END
51
2326
14969
1
FOR SFLSQ3 AND CHLNK3
LIST
LABEL
C
30
31
33
34
35
36
37
38
39
SUBROUTINE PRST
PRINT STATISTICAL RESULTS
COMMON CMA#CMBCMCCMDCME*CMF
EQUIVALENCE (CMATITLE),(CMA(32),SUMDL)
EQUIVALENCE (CMBNPAR),(CMB(2),NVAR),(CMB(3),NCOUNT),(CMB(4),KSEL)
EQUIVALENCE (CMC5)
EQUIVALENCE (CMDNF)
EQUIVALENCE (CME*FOBS)
EQUIVALENCE (CMFSFC).(CMF(2) SMSFC) (CMF(3),SQDL),(CMF(4)tDELTA1)
1,(CMF(5) SUMFO1) (CMF(6),WDL1) '(CMF(7),WFO1),(CMF(8) DELTA2) (CMF(
29),SUMF02)t(CMF(10) WDL2) (CMF(ll),WF02)
DIMENSION CMA(50),CMB(507),CMC(651),CMD(700),CME(419),CMF(50)
DIMENSION TITLE(12)
WRITE OUTPUT TAPE 2,31,TITLE
FORMAT(lH112A6)
WRITE OUTPUT TAPE 2933
FORMAT(79HODISCREPANCY FACTORS BASED ON INPUT PARAMETERS. NUMERAT
lOR
DENOMINATOR
R)
R1=DELTA1/SUMFOl
WDL1=SQRTF(WDLl)
WFO1=SQRTF(WFOl)
WR1=WDL1/WFO1
R2=DELTA2/SUMFO2
WDL2=SQRTF(WDL2)
WF02=SQRTF(WF02)
WR2=WDL2/WFO2
RSUMDL=SQRTF(SUMDL)
TEST=RSUMDL/SQRTF(FLOATF(NCOUNT-NVAR))
WRITE OUTPUT TAPE 2*349DELTAleSUMFOlR1
FORMAT(48HOR FACTOR INCLUDING ZEROS
F10.3t3H
1
F10.3v5H
F5.3)
WRITE OUTPUT TAPE 2,35,DELTA2,SUMFO2,R2
FORMAT(48HOR FACTOR OMITTING ZEROS
F10.393H
1
F10.395H
F5.3)
WRITE OUTPUT TAPE 2,36#WDL1WFOlWR1
FORMAT(48HOWEIGHTED R FACTOR INCLUDING ZEROS
F10.393H
1
F10.3#5H
F5.3)
WRITE OUTPUT TAPE 2937*WDL2#WF02#WR2
FORMAT(48HOWEIGHTED R FACTOR OMITTING ZEROS
F1O.393H
1
F10.3,5H
F5.3)
WRITE OUTPUT TAPE 2938*RSUMDLPNCOUNTtNVARgTEST
FORMAT(48HOSQUARE ROOT (SUM W(OBS-CALC)**2/(M-N))
F10.3#6H
1 SQRT(1491H-I3,3H) F6.3)
WRITE OUTPUT TAPE 2,39,SMSFC
FORMAT(48HOSUM FCAL
F10.3)
RETURN
202
END
LIST
LABEL
*
*
CCHLNK3
SUBROUTINE CHAIN(KXXKZZ)
COMMON CMAtCMBCMCVDIAG#EPS
EQUIVALENCE (CMATITLE),(CMA(13),MODE),(CMA(14),INV),(CMA(15),ISAN
1),(CMA(16) NPU),(CMA(17),NEW),(CMA(22),NU),(CMA(23)tNS),(CMA(24),A
2),(CMA(25),B),(CMA(26),C),(CMA(27),ALPHA),(CMA(28),BETA),(CMA(29),
3GAMMA),(CMA(30),NFOUR),(CMA(31),NCOR),(CMA(32),SUMDL),(CMA(33),KAR
4D),(CMA(34),ID)
EQUIVALENCE (CMB.NPAR),(CMB(2).NVAR),(CMB(3),NCOUNT),(CMB(4),KSEL)
EQUIVALENCE
(CMC'S),(CMC(51),BO),(CMC(52),MF),(CMC(102),X),(CMC
1(152),Y)*(CMC(202),Z), (CMC(252),BllBI),(CMC(302),B22),(CMC(352),
2B33),(CMC(402),B12),(CMC(452),B23),(CMC(502),B13),(CMC(552),SYM),(
3CMC(602),NAME)
EQUIVALENCE (CMC(652),SN)'(CMC(702),BON),(CMC(703),XN),(CMC(753),Y
1N),(CMC(803),ZN),(CMC(853),BINBilN),(CMC(903),B22N),(CMC(953),B33
2N),(CMC(1003),B12N),(CMC(1053),B23N),(CMC(1103),B13N),(CMC(1153),S
3YMN),(CMC(1203),CHG),(CMC(1403),STD)
EQUIVALENCE (CMC(652)#NFA),(CMC(1821)'AR)
DIMENSION CMA(50),CMB(507),CMC(2403)
DIMENSION AR(14969)*V(200),DIAG(200)
DIMENSION TITLE(12),NEW(5)
DIMENSION S(50),MF(50),X(50).Y(50),Z(50),BI(50),B11(50),B22(50),83
13(50)812(50),B23(50),B13(50).SYM(50),NAME(50)
DIMENSION SN(50),XN(50),YN(50),ZN(50),BIN(50),BllN(50),822N(50),B3
13N(50) 812N(50),B23N(50),B13N(50)
DIMENSION SYMN(50)
DIMENSION CHG(200),STD(200),EPS(200)
DIMENSION KSEL(504)
DIMENSION ART(20)
DIMENSION F(20)
ISAN=ISAN
1
NF=NFA
ISING=O
2
N=((NVAR+1)*NVAR)/2
NM=N
400 DO 406 I=1,NVAR
401
DU
402
403
404
405
406
IF(J-I)4049403,404
IF(AR(N))404,56,404
N=N-1
CONTINUE
CONTINUE
5
CALL 5MIl(AK(NM),NVAKelaiNU)
C
6
7
40!
J=leNVAR
IF ISING = 0, MATRIX IS NON-SINGULAR
IF(ISING)52,6t52
DO 12 I=1,NVAR
EP=0.0
IJ=NM-I+1
IJD=NVAR-1
DO 11 J=1,NVAR
203
EP=EP+AR(IJ)*V(J)
IF(J-I)8,9,10
8
IJ=IJ-IJD
IJD=IJD-1
GO TO 11
DIAG(I)=SQRTF(AR(IJ))
9
10
IJ=IJ-1
11
CONTINUE
EPS(I)=EP
12
CONTINUE
1200 IF(NCOR)120,136,120
120 N=NM
FREQUENCY 121(50)9122(25)
121 DO 124 I=1,NVAR
122 DO 123 J=INVAR
AR(N)=AR(N)/(DIAG(I)*DIAG(J))
N=N-1
123 CONTINUE
124 CONTINUE
Il=1
12=20
NV=NM+1
125 WRITE OUTPUT TAPE 2,126,TITLE
126
FORMAT(lHll2A6)
WRITE OUTPUT TAPE 29127
127 FORMAT(31HOPRINTOUT OF CORRELATION MATRIX)
128 DO 135 K=1,25
129 WRITE OUTPUT TAPE 2,130#Il
130 FORMAT(lH 15)
IT=1
FREQUENCY 1310(50)
1310 DO 1315 I=11,12
MT=NV-I
FREQUENCY 1311(0,0,1)
1311
IF(MT)1312,1312,1313
1312 ART(IT)=0.0
GO TO 1314
1313 ART(IT)=AR(MT)
1314 IT=IT+1
1315 CONTINUE
1316 WRITE OUTPUT TAPE 2,132,(ART(IT),IT=1,20)
132 FORMAT lH F5.2,19F6.2)
133
IF(NM-I2)136,136,134
134 I112+1
12=11+19
135 CONTINUE
GO TO 125
136 SUMCH=0.0
DO 13 I=1,NVAR
SUMCH=SUMCH+EPS(I)*V(I)
13
CONTINUE
FIl=SRIlF(UMDL-bUMH)
ERFIT=FIT/SQRTF(FLOATF(NCOUNT-NVAR))
J=1
14
DO
17 I=1,NPAR
204
22
220
IF(KSEL(I))15,16,15
CHG(I)=EPS(J)
STD(I)=DIAG(J)*ERFIT
J=J+1
GO TO 17
CHG(I)=0.O
STD(I)=0.0
CONTINUE
DO 18 J=1NS
SN(J)=S(J)+CHG(J)
CONTINUE
J=NS+1
BON=BO+CHG(J)
J=J+1
DO 22 I=lNU
SYMN(I)=SYM(I)+CHG(J)
J=J+1
XN(I)=X(I)+CHG(J)
J=J+1
YN(I)=Y(I)+CHG(J)
J=J+1
ZN(I)=Z(I)+CHG(J)
J=J+1
GO TO (20,21,21),ISAN
BIN(I)=BI(I)+CHG(J)
J=J+1
GO TO 22
B11N(I)=Bil(I)+CHG(J)
J=J+1
B22N(I)=B22(I)+CHG(J)
J=J+1
B33N(I)=B33(I)+CHG(J)
J=J+1
B12N(I)=B12(I)+CHG(J)
J=J+1
B13N(I)=B13(I)+CHG(J)
J=J+1
B23N(I)=B23(I)+CHG(J)
J=J+1
CONTINUE
WRITE OUTPUT TAPE 2,126,TITLE
23
24
WRITE UUIPUF IAPE 2'24
FORMAT(79HOESTIMATE OF
15
16
17
18
19
20
21
25
26
27
28
WRITE OUIPUF
29
NUMERAT
lOR
DENOMINATOR
R)
WRITE OUTPUT TAPE 2,25,FITNCOUNTNVARERFIT
FORMAT(50HOSQUARE ROOT (SUM W(OBS-CALC)**2/(M-N))
1H SQRT(14,1H-I3#3H) F6.3)
WRITE OUTPUT IAPE 2v26
FORMAT(28HOBASED ON OUTPUT PARAMETERS.)
WRITE OUTPUT TAPE 2v27
OLD
CHANGE
FORMAT(57HOPARAMETER
1R)
DO 30 J=1,NS
1APE
F9.3,6
NEW
ERRO
2,29,J ,(J),CHG(J),5N(J),tlD(J)
FORMAT(14HOSCALE FACTOR
I2,F9.4,3H
F8.4v2H
F8.4,4H
F7.4)
205
30
31
32
33
34
340
35
36
37
38
39
40
41
42
43
44
45
46
47
470
471
472
473
474
CONTINUE
J=NS+1
WRITE OUTPUT TAPE 2,31,BOCHG(J),BONSTD(J)
FORMAT(16HOOVERALL B
F9.4,3XF8.4,2X,F8.4,4XF7.4)
J=J+1
DO 47 I=1,NU
WRITE OUTPUT TAPE 2,33,INAME(I)
FORMAT(6HOATOM 12,3XA6)
WRITE OUTPUT TAPE 2,349MF(I)
FORMAT(19H FORM FACTOR
12)
WRITE OUTPUT TAPE 2,340,SYM(I),CHG(J),SYMN(I),STD(J)
FORMAT(15H ATOM S.F.
4(F1O,7,1X))
J=J+1
WRITE OUTPUT TAPE 2,35,X(I),CHG(J),XN(I),STD(J)
FORMAT(15H X
4(F1O.7,1X))
J=J+1
WRITE OUTPUT TAPE 2,36,Y(I),CHG(J),YN(I),STD(J)
FORMAT(15H Y
4(F10.7,1X))
J=J+1
WRITE OUTPUT TAPE 2,37,Z(I),CHG(J),ZN(I),STD(J)
FORMAT(15H Z
4(F1O.7#1X))
J=J+1
GO TO (38,40,40),ISAN
WRITE OUTPUT TAPE 2,39,BI(I),CHG(J),BIN(I),STD(J)
FORMAT(15H ATOMIC B
4(F1O.7,1X))
J=J+1
GO TO 47
WRITE OUTPUT TAPE 2,41,Bll(I),CHG(J),BilN(I),STD(J)
FORMAT(15H BETA(1,1)
4(F1O.7,1X))
J=J+1
WRITE OUTPUT TAPE 2,42,B22(I),CHG(J),B22N(I),STD(J)
FORMAT(15H BETA(2t2)
4(F1O.7,1X))
J=J+1
WRITE OUTPUT TAPE 2,43,B33(I),CHG(J),B33N(I),STD(J)
FORMAT(15H BETA(3,3)
4(F1O.7,1X))
J=J+1
WRITE OUTPUT TAPE 2,44,B12(I),CHG(J),B12N(I),STD(J)
FORMAT(15H BETA(1,2)
4(F1O.7,1X))
J=J+1
WRITE OUIPUT TAPE 2,45,B13(I),CHG(J),B13N(I),5iD(J)
FORMAT(15H BETA(1,3)
4(F1O.7,1X))
J=J+1
WRITE OUTPUT TAPE 2,46,B23(I),CHG(J)9823N(I),STD(J)
FORMAT(15H BETA(2,3)
4(F10./91X))
J=J+1
CONTINUE
CALL TEST(KA)
WRITE IAPE 10,NtABCALPHABIAGAMMASNBONMFXNYN,4NBllNB2
12NB33NB12NBl3NB23NSYMNAMENPARNVARKSEL
READ TAPE 7,MHMK'MLFOBS'SIGMAMS1,EXT1,EXT2,LASTRHORHOSQF
WRITE TAPE 10,MH'MK'MLFOBSSIGMAM,1,EXT1,EXT2,LASTRHORHOSQF
IF(LAST)472,470,472
IF(KARD)473,474,473
CALL CARD
IF,(ID)475,476,475
206
475
476
CALL DIST
REWIND 10
REWIND 7
IF(KA)51948,51
48
49
50
51
54
55
56
57
58
*
*
READ INPUT TAPE 4,49,IEND
FORMAT(71X.Il)
IF(IEND)50t51,50
RETURN
CALL EXIT
WRITE OUTPUT TAPE 2,53
FORMAT(19HOMATRIX IS SINGULAR)
WRITE OUTPUT TAPE 2.54
FORMAT(39HOLISTING OF DIAGONAL OF INVERTED MATRIX)
WRITE OUTPUT TAPE 2,55,(DIAG(I),I=1,NVAR)
FORMAT(10E10.3)
GO TO 58
WRITE OUTPUT TAPE 2953
WRITE OUTPUT TAPE 2,57%1
FORMAT(22HODIAGUNAL TERM NUMBER 13,38H IN THE NORMIAL EQUATION MATN
lIX IS ZERO)
ISING=1
GO TO 404
CALL EXIT
END
LIST
LABEL
C
1
3
SUBROUTINE TEST(K)
TEST FOR NEGATIVE TEMPERATURE FACTORS
COMMON CMACMBCMC
EQUIVALENCE (CMATITLE),(CMA(13),MODE),(CMA(14),INV),(CMA(15),ISAN
1),(CMA(16),NPU), (CMA(17),NEW),(CMA(22),NU),(CMA(23),NS),'(CMA(24),A
2),(CMA(25),B),(CMA(26),C),(CMA(27),ALPHA),(CMA(28),BETA),(CMA(29),
3GAMMA),(CMA(30),NFOUR),(CMA(31),NCOR),(CMA(32),SUMDL),(CMA(33),KAR
4D),(CMA(34),ID)
EQUIVALENCE (CMB#NPAR)
EQUIVALENCE
(CMC'S),(CMC(51),BO),(CMC(52),MF),(CMC(102),X),(CMC
1(152),Y)'(CMC(202),Z), (CMC(252),Bll BI),(CMC(302),B22),(CMC(352),
2B33),(CMC(402),B12),(CMC(452),B23),(CMC(502),B13),(CMC(552), YM), (
3CMC(602)#NAME)
EQUIVALENCE (CMC1652),5N),(CMC( /U2),bUN)9(CML 7U3),AN),(CMC( l53),Y
1N),(CMC(803),ZN),(CMC(853),BINBilN),(CMC(903),B22N),(CMC(953),B33
2N),(CMC(1003) l812N),(CMC(1053),623N),(CMC(1103),Al3N),(CMC(1603),C
3HG),(CMC(2103),STD),(CMC(2203),EPS)
DIMENSION CMA(50),CMB(507),CMC(2403)
DIMENSION S(50),MF(50),X(50),Y(50),Z(50),BI(50),B11(50),B22(50),B3
13(50 )B12(50),B23(50),B13( 50) ,SYM(5U),NAME(50)
DIMENSION SN(50),XN(50),YN(50),ZN(50),BIN(50),BllN(50),B22N(50),B3
13N(50),B12N(50),B23N(50),B13N(50)
K=0
ISAN=ISAN
GO TO(3,5,5),ISAN
DU 4 I1,NU
BI(I)=BIN(I)+BON
207
4
5
CONTINUE
GO TO 7
AA=BON*A*A/4.0
BB=BON*B*B/4.O
CC=BON*C*C/4*0
AB=BON*A*B*COSF(GAMMA)/4.0
BC=BON*B*C*COSF(ALPHA)/4.0
AC=BON*A*C*COSF(BETA)/4.0
DO 6 I=lNU
Bll(I)=B11N(I)+AA
B22(1)=B22N(I)+BB
B33(I)=B33N(I)+CC
B12(I)=B12N(I)+AB
B23(I)=B23N(I)+BC
B13(I)=B13N(I)+AC
6
CONTINUE
7
GO TO(8,12,12),ISAN
8
DO 11 I=1,NU
IF(BI(I))9,11,11
9
WRITE OUTPUT TAPE 2,10,1
10
FORMAT(28HOTEMPERATURE FACTOR OF ATOM I292-H IS NOT POSiTIVt-OLFIN
1ITE)
BIN(I)=.o1
11
CONTINUE
GO TO 20
12
DO 19 I=1,NU
13
IF(811(1))18.14,14
14
IF(B22(I))18,15,15
15
IF(B33(I))18.16,16
16
IF(Bil(I)*B22(I)*B33(I)+2.0*B12(I)*B23(I)*B13(I)-(B13(I)*B13(I)*B2
12(I)+B12(I)*B12(I)*B33(I)+Bll(I)*B23(I)*B23(I)))18,160,160
160
IF(B22(I)*633(I)-B23(I)**2)18,161,161
161
IF(B11(I)*B33(I)-Bl3(I)**2)18,162,162
162
IF(Bll( )*B22(I)-B12(I)**2)18.17,17
17
GO TO 19
18
WRITE OUTPUT TAPE 2,10,1
K=K+1
19
CONTINUE
RETURN
20
END
*
*
C
LIST8
LABEL
SUBROUTINE CARD
PUNCH ATOM PARAMETER CARDS
COMMON CMA#CMB#CMCVPDIAGPEPS
EQUIVALENCL (CMATIILL),(CMA(13),MUDE),(CMA(14),INV),(CMA(15),IAN
1),(CMA(16),NPU),(CMA(17),NEW),(CMA(22),NU),(CMA(23),NS),(CMA(24),A
2),(CMA(25),B),(CMA(26),C)'(CMA(27),ALPHA),(CMA(28),BEIA),(CMA(29),
3GAMMA),(CMA(30),NFOUR),(CMA(31),NCOR),(CMA(32)SUMDL)(CMA(33)KAR
4D),(CMA(34),ID)
EQUIVALENCE (CMBNPAR),(CMB(2),NVAR),(CMB(3),NCOUNT),(CMB(4),KSEL)
EQUIVALENCL
(CMC'5,tMC(51),BO)(MC(52),),MC(IU),),1cML
1(lp2),Y)*(CMC(202)vZ)
(CMC(252).BllBI),(CMC(302)9B22),(CMC(352)9
208
2B33),(CMC(402),B12),(CMC(452),B23),( CMC(502),B13),(CMC(552),SYM),(
3CMC(602) ,NAME)
EQUI-VALENCE (CMC(652),SN) '(CMC(702), BON)9(CMC(703),XN)f(CMC(753),Y
1N),(CMC(803),ZN),(CMC(853),BINBilN) ,(CMC(903),B22N),(CMC(953),B33
2N),(CMC(1003),B12N),(CMC(1053),B23N) ,(CMC(1103),B13N),(CMC(1603),C
3HG),(CMC(2103),STD)
EQUIVALENCE (AR9CMC(652),NFA)
EQUIVALENCE (CMC(1153),SYMN)
DIMENSION CMA(50),CMB(507),CMC(2403)
DIMENSION AR(14969).,V(200) ,DIAG(200)
DIMENSION TITLE(12),NEW(5)
DIMENSION S(50),MF(50),X(50),Y(50),L(50),BI(50) '811(50),B22(50),83
13(50),B12(50),B23(50),813(50),SYM(50),NAME(50)
DIMENSION SN(50),XN(50) YN(50) ,N(50),BIN(50),BllN(50),B22N(50)B3
13N(50),B12N(50),B23N(50),B13N(50)
DIMENSION SYMN(50)
DIMENSION CHG(200),STD(200),EPS(200)
DIMENSION DF(200)
DIMENSION KSEL(504)
DIMENSION ART(20)
DIMENSION F(20)
ISAN=ISAN
1 DO 3 I=1,NS
PUNCH 2#SN(I)
2 FORMAT (F9.4)
3 CONTINUE
PUNCH 4,BON
4 FORMAT (F9.4)
5 DO 11 I=1,NU
6 GO TO (7,9,9),ISAN
7 PUNCH 8,MF(I),XN(I),YN(I),ZN(I),BIN(I)
8 FORMAT (I2,5X#3F7.59F7.4)GO TO 11
9 PUNCH 10,MF(I),XN(I)*YN(I),2N(I),B11N(I),B22N(I),B33N(I),B12N(I),B
X13N(I),B23N(I)
10 FORMAT (1295X67.5,3/.4)
11 CONTINUE
12 END FILE 3
13 RETURN
END
FAP
*
DIST
COUNT
ENTRY
REM
TRA
END
2
DIST
DUMMY SUBROUTINE DIST
1,4
-
C
Appendix II
Program for computation of drilling coordinates
for crystal structure models
Chapter III gives a mathematical account of the procedures
for calculating drilling coordinates for crystal structure models.
This contains some details of a program written for the IBM 7090
computer which will compute these drilling coordinates.
written, the program is a rather elementary one,
As it is
since it contains
no provision for generating coordinates of symmetry-related atoms.
It works well, however, and its use not only saves time in calculation
of drilling coordinates, but it reduces the possibility of computational
errors which could cause considerable trouble in putting a model
together.
Since the computational methods are very similar, an obvious
extension of this program would be to compute interbond angles as
well as drilling coordinates and interatomic distances.
In such a
program, symmetry transformations could be included so that only
the coordinates of atoms of the asymmetric unit would have to be
submitted with the program.
In addition, although they are not
required for the drilling coordinates, the errors in interatomic
and interbond angles could also be computed.
Directions for running the program DRILL
This program computes the driling angles
atoms N. coordinating atom N,,
4
and p for all
given cell parameters,
atom coordL-
nates, and information defining coordination polyhedra.
No symmetry
transformations are performed on input atom coordinates.
Therefore
all transformations must be performed by the program user and all
atoms with unique coordinates must be supplied to the program.
The
program is designed to be run under the Fortran Monitor System and
is coded entirely in FORTRAN.
DATA CARDS FOR DRILL
1.
TITLE CARD. Format (12A6) Cols. 1-72; Any identification
information may be included.
2.
CELL PARAMETER CARD.
cols.
Format (3(F7.4, 3X), 3(F6.2,4X), 12)
Format
Parameter
1-7
F7. 4
11-17
i
21-27
31-36
alpha (in degrees)
F6.4
41-46
beta (in degrees)
i
51-56
gamma (in degrees)
t
61-62
NT
12
NT is the number of atoms and is equal to the
number of atom parameter cards or atom coordination cards.
NT must not be greater than 72.
0
L"
3.
ATOM PARAMETER CARDS. Format (12, 2X,A6, 2X, 3(12, 2X), 3F6.4)
Cols.
Parameter
1-2
atom number, n.
12
5-10
atom name
A6
13-14
no. of atoms coordinating atom n 12
L
17-18
no. of the atom with rho = 0.0
"
21-22
no. of the atom with phi = 0. 0
"
25-30
x.
F6.4
31-36
yI
37-42
Z.
Format
L
L
t
L
Columns 13-14, 17-18, and 21-22 are left blank if no atoms
coordinate atom n..
(i.e. Atom n. is only coordinated to some
other atom in this case.)
x, y, and z may be negative or greater than 1.0
There are NT such cards, one card for each atom with
unique coordinates.
4.
ATOM COORDINATION CARDS. Format(721)
NT cards, one for each atom.
These cards designate the
numbers of the atoms which coordinate atom a..
-L
A blank
card must be used if no atoms coordinate atom n,.
Cols. 1-72: Punch 1 in the columns whose numbers correspond
to the atom numbers coordinating atom n.
Cards of type 3 and 4 are arranged such that n. increases uniformly.
The two cards of types 3 and 4 for atom n. are placed together with
that of type 3 first.
DESCRIPTION OF DATA DECK
*
DATA
Title card
Cell parameter card
Atom parameter card for atom I
Atom coordination card for atom i
Atom parameter card for atom 2
Atom coordination card for atom 2
Atom parameter card for atom NT
Atom coordination card for atom NT
DESCRIPTION OF OUTPUT
The numbers, names and angles phi. and rho of all atoms coordinating
some atom n. are printed out for each coordination group and its
-L
enantiomorphic equivalent, i. e.,
for atoms related by a mirror plane
or an inversion center.
In addition, interatomic distances are printed out and may be used
as a check on the correctness of the input data.
213
FORTRAN LISTING OF PROGRAM DRILL
C
1
2
3
4
ROUTINE TO COMPUTE DRILLING COORDINATES PHI AND RHO + INT DIST
DIMENSION TITLE(12),N(72),NAME(72),NCRD(72),NRHO(72),NPHI(72),
1X(72),Y(72),Z(72),NC(72,72),DX(72),DY(72),DZ.(72),DIST(72),RHO(72)
2ARHO(72),U(72),V(72),W(72),SIZE(72),PHI(72)$PHIN(72),XP(72),YP(72)
3,ZP(72),T(3,3)
FREQUENCY 11(0,3,1),13(0,10,1),23(01O ,1),31(0,101 ),33(5,1,5)
READ INPUT TAPE 4,1,(TITLE(I),I=1,12)
FORMAT(12A6)
WRITE OUTPUT TAPE 2,2,(TITLE(I),I=1,12)
FORMAT(lH112A6)
WRITE OUTPUT TAPE 2,3
FORMAT(88HOCENTER ATOM
COORDINATING ATOM
PHI
RHO *
PHI
RHO)
ENANTIOMORPH
1
READ INPUT TAPE 4,4,A,B,C,ALPHABETAGAMMANT
FORMAT(3(F7.4,3X),3(F6.2,4X),12)
DO 7 I=1,NT
READ INPUT TAPE 4,5,N(I),NAME(I),NCRD(I),NRHO(I),NPHI(I),X(I),Y(I)
1,Z(I)
5
6
7
FORMAT(I2,2XA6,2X,3(I2,2X),3F6.4)
READ INPUT TAPE 4,6'(NC(IoJ)qJ=1,72)
FORMAT(7211)
CONTINUE
PI =3.1415927
RAD =PI/180.0
ALPHA=ALPHA*RAD
BETA=BETA*RAD
GAMMA=GAMMA*RAD
ONE=(COSF(ALPHA)-COSF(BETA)*COSF(GAMMA))/SINF(GAMMA)
T(1,1)=A
T(1,2)=B*COSF(GAMMA)
T(1,3)=C*COSF(BETA)
T(2,1)=0.0
T(2,2)=B*SINF(GAMMA)
T(2,3)=C*ONE
T(3,1)=0.0
T(3,2 )=0.0
T(3,3)=C*SQRTF(SINF(BETA)**2-ONE**2)
DO 8 I=1,NT
XP(I)=X(I)*T(1,1)+Y(I)*T(1,2)+Z(I)*T(1,3)
YP(I)=X(I)*T(2,1)+Y(I)*T(2,2)+Z(I)*T(2,3)
ZP(I)=X(I)*T(391)+Y(I)*T(3,2)+Z(I)*T(3,3)
8
10
11
12
121
13
14
CONTINUE
DO 50 I=1,NT
IF(NCRD(I))50,50,12
WRITE OUTPUT TAPE 2,121,N(I),NAME(I)
FORMAT(lHOI2,2XA6)
DO 21 J=1,72
IF(NC(IoJ))21,21,14
DX(J)=XP(I)-XP(J)
DY(J)=YP(I)-YP(J)
19
21
22
23
24
241
242
243
244
245
246
28
29
30
31
32
321
322
323
324
329
33
35
36
41
49
50
214
DZ(J)=ZP(I)-ZP(J)
DIST(J)=DX(J)**2+DY(J)**2+DZ(J)**2
DIST(J)=SQRTF(DIST(J))
CONTINUE
DO 28 J=1,72
IF(NC(ItJ))28,28,24
K=NRHO(I)
SCALA=DX(K)*DX(J)+DY(K)*DY(J)+DZ(K)*DZ(J)
RHC(J)=SCALA/(DIST(K)*DIST(J))
IF(RHO(J)-1.0)243,242,242
RHO(J)=0.0
GO TO 246
IF(RHO(J)+1.0)244,244,245
RHO(J)=3ol415927
GO TO 246
RHO(J)=ACOSF(RHO(J))
ARHO(J)=RHO(J)/RAD
U(J)=(DY(K)*DZ(J))-(DZ(K)*DY(J))
V(J)=(DZ(K)*DX(J))-(DX(K)*DZ(J))
W(J)=(DX(K)*DY(J))-(DY(K)*DX(J))
SIZE(J)=SINF(RHO(J))*DIST(K)*DIST(J)
CONTINUE
L=NPHI(I)
UU=(DY(K)*W(L))-(DZ(K)*V(L))
VV=(DZ(K)*U(L))-(DX(K)*W(L))
WW=(DX(K)*V(L))-(DY(K)*U(L))
DO 49 J=1,72
IF(NC(ItJ))49,49,32
SCALA=U(L)*U(J)+V(L)*V(J)+W(L)*W(J)
PHI(J)=SCALA/(SIZE(L)*SIZE(J))
IF(PHI(J)-1.0)323,322,322
PHI(J)=O0*
GO TO 36
IF(PHI(J)+1.0)324,3249329
PHI(J)=180.0
GO TO 36
PHI(J)=ACOSF(PHI(J))
PHI(J)=PHI(J)/RAD
SCALE=UU*U(J)+VV*V(J)+WW*W(J)
IF(SCALE)35,36,36
PHI(J)=360.0-PHI(J)
PHIN(J)=360.0-PHI(J)
TAPE 2,41iJNAME(J),PHI(J),ARHO(J),PHIN(J),ARHO(J),
WRITE OUTPUT
1DIST(J)
FORMAT(19XI2,2XA6,6XF6.1,3XF6.1,24XF6.1,3XF6,1,5XF7.3)
CONTINUE
CONTINUE
CALL EXIT
END
Appendix III
Computation of diffractometer settings
In recent years many laboratories engaged in single-crystal
x-ray diffraction work have begun to use counter diffractometers to
measure diffraction intensities.
Many of the difficulties which were
first encountered in this area have been overcome, and, with the
advent of complete automation of the equipment, this method should
be much better than the conventional film methods.
Prewitt (1960)
published the relations needed to compute the angular settings for
the equi-inclination Weissenberg diffractometer (Buerger, 1960)
which is being used in a number of laboratories in the U.
abroad.
S. A. and
This appendix describes an IBM 7090 program written to
compute these settings.
Program description.
This program, which is designed to
operate under the Fortran Monitor System, consists of a main program
DFSET and several subroutines.
After a brief description of each
section of the program, instructions for setting up a run will be given.
DFSET.
The main program reads the input data from A2,
computes necessary constants, writes headings on the output tape A3
and transfers to the selected hki generation subroutine INDEXi or
INDEXZ.
Upon return from one or the other of these subroutines, a
selected hkl has been stored in COMMON.
The main program
computes s inD and compares it to a maximum allowable sin9 prescribed
by the input data.
If the computed sin B is too large, one of the
indexing routines is called again and a new hki
generated; otherwise
the main program goes on to compute T, 4,, and 1/Lp as given by
Prewitt (1960).
coordinates,
Also computed are the precision Weissenberg
and , (Buerger, 1942; Prewitt, 1960), which are very
useful in indexing precision Weissenberg films.
OUTPUT is then
called to write tapes for printing and/or punching.
from OUTPUT,
Upon the return
the main program again calls one of the indexing
routines.
INDEXI.
data deck.
This subroutine will read hklts from cards in the
Upon reading a termination card as described below, the
subroutine TERM will be called.
INDEX2.
This routine is coded in FAP and generates hklts
according to extinction rules given in International Tables (1952),
Vol. 1, pp. 53-54.
Input data to the program contains lower and
upper limits for hkl and INDEX2 examines all combinations between
these limits for compliance with the extinction rules.
If a particular
hki passes, it is stored in COMMON and control is returned to the
main program.
INDEX2 could be used in conjunction with any program which
requires the generation of hkls according to a particular extinction
rule.
All communication with the main program is through COMMON.
The information needed by INDEX2 to operate successfully is found
at the end of the program listing in the COMMON section.
These
217
symbols are defined in the instructions for running the program.
Upon exceeding the upper limits of hkl,
to terminate the run.
exceeded,
TERM will be called
If for some reason h, k, or I is accidentally
sense light 1 will be turned on before TERM is called.
OUTPUT. h, k, 1,2T , c,
to the calling sequence)
and
I,
1/n, sin 0, and (with their addition
are written on A3 and/or B4 for
*
printing and/or punching, respectively.
Note that statement 24 is a
PUNCH statement which might have to be changed if this would cause
on-l1ne punching at a particular computing center.
MIFLIP.
An M. I. T. subroutine which writes punched output
identification records on B4.
May be eliminated by removing state-
ments 131 and 131-1 in the main program.
TERM. Depending on the end of run indicators, this routine
will either call EXIT and stop the run or will call CHAIN (2, A4)
which will cause the program to be read in from A4 and a new set
of data evaluated.
Speed of computation. Settings are computed at a rate of 30-45
per second, depending on the extinction rules and the speed of the tape
drives being used.
The punched cards produced by this program may be used as input
to the Burnham (1961) program DTRDA.
Running the program
Data deck. The cards of the data deck are assembled in the
following order:
FORMA T
1.
TITLE:
Any identification in cols. 1-72.
12A6
2.
FLP:
Information for identifying punched
4A 6
output.
May be blank if no punched
output is requested..
Col.
3.
1-6
Problem number (at M. I.T.)
7-12
Programmer number
13-24
Blank
Sense card
Col.
1
LSQ:
I
-
cell constants are to be
I
read from data cards
2
-
if this is a chain link with
the Burnham (1961) lattice
parameter least-squares program.
IKL:
1
-
hkl are to be read from data 11
cards
2
-
internal generation of hki
is to be used
NPRNTR
-1
2-
print output
11
punch output
16-20 SINTHO: upper limit of sin 1 for which
F5.4
settings are to be computed.
When computing for all reflections,
this probably should be . 99999
o-~
rather than 1.0 because of
possible trouble in the arcsin
routine.
23-25
MH1:
lower limit for h
28-30
MH2:
upper
"1
I
I
33-35
MKI:
lower
It
"
k
38-40
MK2:
upper
"
II
"t
43-45
MLI:
lower
"
"I
48-50
ML2:
upper
"
U "
53
ISE T:
12
0
LAST:
It
-
I
-
I
rotation axis is c
3
72
13
t
I
I1
b
"a
ifthis is the last (or only)
H
group of settings to be
computed
1
-
if there is another set of
data and chain cards are in
their proper places
4.
Reciprocal cell data and space group extinction selection.
The cell data is not needed if the program is operated with the
Burnham lattice constant refi nement program.
Col.
1-7
Al:
a* (1/a, not X/a)
8-14
BI:
b*
15-21
Cl:
c*
22-28
ALRHAI:
a* (in degrees)
29-35
BE TAI:
p*
36-42
GAMMAI:
y*
43-49
WVLNG:
x (.)
F7. 6
F7.4
0
220),
61-69
Extinction rules
61
LHKL:
1-
h+k= 2n
2 -
k + I
3 -
62
63
64
65
LHKO:
LOKL:
LHOL:
LHHL:
+h =2n
hkl all odd or all even
5 -
h+k+I= 2n
6 -
h+k+1= 3n
7-
h-k+1= 3n
8 -
h - k = 3n
0 -
no restrictions
1
-
h = 2n
2-
k = 2n
3-
h + k = 2n
4-
h + k = 4n
1 -
h = 2n
2-
1 = 2n
3-
k + 1 = 2n
4-
k + 1 = 4n
1
-
I = 2n
2-
h = 2n
3-
1 + h = 2n
4-
1 + h = 4n
1
1 = 2n
-
2h +1 = 4n
66
LHML:
1 - 1 = 2n
67
LHOO:
1-
h = 2n
2-
h = 4n
1
k = 2n
LHOKO:
2n
4 -
2 -
68
911
-
(hhl)
FT
69
5.
LOOL:
1-
I= 2n
2 -
I = 3n
3 -
1 = 4n
4 -
1 = 6n
hkl cards--needed only if there is a I in col. 4 of the sense
card.
If this option is used, diffractometer settings will be com-
puted for all reflections for which sin 6 is less than the allowable
maximum.
Col.
1-3
MH:
h
5-7
MK:
k
9-11
ML:
1
72
IEOF:
End of hkl deck indicator.
13
Ii
Blank except on last card
which is blank except for
I in col. 72.
The deck should be composed as follows:
*
I.D. card
*
XEQ card
*
CHAIN(2,A4) -- include this card if more than one
set of data is to be run and col. 72 of the same card
contains a one.
Also include this card if the program
is being run as a chain link with the Burnham (1961)
lattice constant refinement program.
DFSET (main program)
Subroutine s
*
DATA
Data deck for 1st set of reflections
"1
I
t
2nd
It
Data deck for nth set of reflections
References
Buerger, M. J. (1942),
X-Ray Crystallography. John Wiley and Sons,
New York.
Buerger, M. J. (1960),
Crystal Structure Analysis.
John Wiley and
Sons, New York.
Burnham, Charles W. (1961),
The Structures and Crystal Chemistry
9fthe Aluminum-silicate Minerals.
Ph.D. thesis, M. I. T. ,
Cambridge, Mass.
Prewitt, Charles T. (1960),
The parameters T and < for equi-
inclination, with application to the single-crystal diffractometer.
Z. Krist. 114, 355-360.
223
LIST8
*
LABEL
CDFSET
C
PROGRAM FOR COMPUTING EQUI-INCLINATION SINGLE-CRYSTAL
1/11/62
DIFFRACTOMETER SETTINGS PLUS 1/LP AND SIN THETA
C
COMMON AlB1,ClALPHA1,BETA1,GAMMAlNTEST,
IMHMKMLLASTNPRNTMH1,MH2,MK1,MK2'ML1,ML2,ISET,
2LHKLLHKOLOKLLHOLLHHL.LHMLLHOO.LOKOLOOL
FREQUENCY 25(1,50,1),28(5v0,1)o31(8,0,1)
DIMENSION FLP(4),TITLE(12)
50
READ INPUT TAPE 4,1,(TITLE(I),I=1,12)
FORMAT(12A6)
1
READ INPUT TAPE 4,2,FLP(3),FLP(4),FLP(1),FLP(2)
2
FORMAT(4A6)
READ INPUT TAPE 4,4,LSQIKLoNPRNTSINTHOMH1,MH2,MK1,MK2,ML1,ML2,
3
1ISETLAST
FORMAT(Il,2XIl,2XIl,8XF5.4,2X,6(I3,2X),Il118XII)
4
41
GO TO(5,7),LSQ
READ INPUT TAPE 4,6,AlB1ClALPHA1,BETA1,GAMMAlWVLNG'LHKLLHKOL
5
1OKLLHOLLHHLLHMLLHO0,LOK0,LOOL
6
FORMAT(3F7.6,3F7.3,F7.4,11X,9Il)
NTEST=1
GO TO 9
READ INPUT TAPE 4,8,WVLNGLHKLLHKOLOKLLHOLLHHLLHMLLHO0,LOKO,
7
1LOOL
FORMAT(42XF7.4,11X,9Il)
8
IF(NTEST-1)35,9,35
81
9
PI=3.1415927
RAD=PI/180.0
GO TO(101112),ISET
10
A=Al*WVLNG
B=Bl*WVLNG
C=Cl*WVLNG
ALPHA=ALPHA1*RAD
BETA=BETA1*RAD
GAMMA=GAMMA1*RAD
GO TO 13
A=Cl*WVLNG
11
B=Al*WVLNG
C=B1*WVLNG
ALPHA=GAMMA1*RAD
BETA=ALPHA1*RAD
GAMMA=BETA1*RAD
GO TO 13
A=B1*WVLNG
12
B=Cl*WVLNG
C=Al*WVLNG
ALPHA=BETA1*RAD
BETA=GAMMA1*RAD
GAMMA=ALPHA1*RAD
COSA=COSF(ALPHA)
13
COSB=COSF(BETA)
COSG=COSF(GAMMA)
SING=SINF(GAMMA)
ABG=A*B*COSG
~~~~~1
21JL
BCA=B*C*COSA
CAB=C*A*COSB
AA=A*A
BB=B*B
CC=C*C
BCOSG=B*COSG
CCOSB=C*COSB
BSING=B*SING
CNST1=(COSA*COSA+COSB*COSB-2.0*COSA*COSB*COSG)/(SING*SING)
CNST2=CNST1/4.0
CNST3=C*(COSA-COSB*COSG)/SING
TLTST=100
GO TO(14,131,131),NPRNT
131 CALL PILF1(FLP,4)
GO TO(141,18,141),NPRNT
14
WRITE OUTPUT TAPE 2,142t(TITLE(I),I=1,12)
141
FORMAT(lHll2A6)
142
WRITE OUTPUT TAPE 2,15,AlB1,C1,WVLNG
LAMBDA = F7.
B* = F7.6,7H C* = F7.6,11H
FORMAT(7H A* = F7.6,7H
15
14)
WRITE OUTPUT TAPE 2,16,ALPHAlBETA1GAMMAl
FORMAT(11H ALPHA* = F8.4,10H BETA* = F8.4.11H GAMMA* = F8.4)
16
WRITE OUTPUT TAPE 2,17
SIN
1/ LP
PHI
UPSILON
L
H
K
FORMAT(63HO
17
1(THETA))
18
181
19
20
21
22
23
24
25
26
27
28
29
DO 34 I=1950
GO TO(19,20),IKL
CALL INDEXl
GO TO 21
CALL INDEX2
GO TO(22,23,24),ISET
TH=FLOATF(MH)
TK=FLOATF(MK)
TL=FLOATF(ML)
GO TO 25
TH=FLOATF(ML)
TK=FLOATF(MH)
TL=FLOATF(MK)
GO TO 25
TH=FLOATF(MK)
TK=FLOATF(ML)
TL=FLOATF(MH)
IF(TL-TLTST)26,27,26
TLTST=TL
TLLCC=TL*TL*CC
SIG=TH*TH*AA+TK*TK*BB+2.0*(TH*TK*ABG+TK*TL*BCA+TL*TH*CAB)
SINTH=(SQRTF(SIG+TLLCC))/2.0
IF(SINTH-SINTHO)29,29,181
XISQ=SIG+TLLCC*CNST1
XI=SQRTF(XISQ)
RSQ=1.0-TLLCC*(.25-CNST2)
OMEGA=ASINF(SQRTF(XISQ/(4.0*RSQ)))
XIX=TH*A+TK*BCOSG+TL*CCOSB
XIY=TK*BSING+TL*CNST3
FREQUENCY 291(50,1,50),292(1,0,1)
291
292
293
294
295
296
30
31
32
33
34
35
36
37
IF(XIX)2959292,295
IF(XIY)293,295,294
PSI=-PI/2.0
XIX=1.0
GO TO 296
PSI=PI/2.0
XIX=1.0
GO TO 296
PSI=ATANF(XIY/XIX)
PHI=PI-(XIX/ABSF(XIX))*PI/2.0-PSI+OMEGA
UPSILN=2oO*OMEGA
VLP=(2.0*RSQ*SINF(UPSILN))/(1.0+(COSF(2.0*ASINF(SINTH)))**2)
PHI=PHI/RAD
UPSILN=UPSILN/RAD
PSI=PSI/RAD
IF(PHI-360.0)33#32,32
PHI=PHI-360.0
GO TO 31
CALL OUTPUT(UPSILNPHIVLPtSINTH)
CONTINUE
GO TO 14
WRITE OUTPUT TAPE 2936
TERMINATE.)
FORMAT(4lHlLEAST-SQUARES REFINEMENT BAD#
CALL EXIT
END
LIST8
LABEL
*
*
21
22
23
24
25
26
27
SUBROUTINE OUTPUT(UPSILNPHIVLPSINTH)
COMMON A1B1,ClALPHA1,BETA1GAMMA1'NTEST,
IMHMKtMLLASTNPRNT
GO TO(22,23,22),NPRNT
WRITE OUTPUT TAPE 29 269 MHMKMLUPSILNPHIVLPSINTH
GO TO(25*24,24),NPRNT
PUNCH 27,MHMKMLUPSILNPHIVLP.SINTH
RETURN
I3,2X#I3,2XI3,2F12.2,F12.49F11.4)
FORMAT(2H
FORMAT(3I3,2X,2(F6.2,1X),2(F6.4,1X))
END
LIST8
LABEL
*
*
C
1
2
3
4
5
SUBROUTINE INDEX1
SUBROUTINE FOR READING HKL FROM DATA CARDS
COMMON AlB1,ClALPHA1,BETAlGAMMA1NTEST,
1MHMKMLLAST
READ INPUT TAPE 4,2,MHMKMLtIEOF
FORMAT(I3,1X,I3,1XI3,59XIl)
IF(IEOF)5,4,5
RETURN
CALL TERM
END
226
*
*
1
2
3
4
5
6
7
8
9
10
*
LIST8
LABEL
SUBROUTINE TERM
COMMON AlB1,ClALPHA1,BETA1,GAMMA1NTEST,
1MHMKMLLASTNPRNT
IF(SENSE LIGHT 1)9,2
GO TO(4,3,3),NPRNT
END FILE 3
IF(LAST)5,6,5
CALL CHAIN(2#A4)
WRITE OUTPUT TAPE 2,7
FORMAT(11HOEND OF RUN)
CALL EXIT
WRITE OUTPUT TAPE 2,10
FORMAT(116H1AN ERROR HAS OCCURRED IN THE INDEXING SUBROUTINE. ONE
1OF THE INDICES HAS EXCEEDED ITS STATED LIMIT. RUN TERMINATED.)
GO TO 8
END
FAP
COUNT
ENTRY
REM
REM
INDEX2 SXD
SXA
NOP
CLA
STO
CLA
STO
CLA
STO
CLA
STO
REM
NOEXT ZET
TRA
ZET
TRA
ZET
TRA
ZET
TRA
ZET
TRA
ZET
TRA
ZET
TRA
ZET
TRA
ZET
405
INDEX2
SUBROUTINE TO GENERATE HKL AND TO REJECT HKL WHICH DO NOT
CONFORM TO SPACE GROUP EXTINCTION RULES.
SAVEo1
SAVE92
TRABA
*-2
MHl
MH
ENTER INITIAL INDICES
MK1
ML1
ML
CHECK FOR ZEROES IN EXTINCTION RULE REQUESTS
LHKL
AA
LHKO
AA
LOKL
AA
LHOL
AA
LHHL
AA
LHML
AA
LHOO
AA
LOKO
AA
LOOL
227
AA
AB
AC
AD
AE
AF
AG
AH
AK
TRA
CLA
STO
TRA
REM
CLA
TZE
PDX
CLA
STO
CLA
TZE
PDX
CLA
STO
CLA
TZE
PDX
CLA
STO
CLA
TZE
PDX
CLA
STO
CLA
TZE
PDX
CLA
STO
CLA
TZE
PDX
CLA
STO
CLA
TZE
PDX
CLA
STO
CLA
TZE
CLA
STO
CLA
TZE
PDX
CLA
STO
AA
BYPASS EXTINCTION RULES
TRAEA
TRNS
AM
CHOOSE EXTINCTION RULES
HKL
LOOL
AB
0,1
XA+4,1
TRNS
LOKO
AC
O1
XB+2,1
TRNS+1
LHOO
AD
0,1
XC+2,1
TRNS+2
LHOL
AE
0,1
XD+4,1
TRNS+3
LOKL
AF
o1
XE+4, 1
TRNS+4
LHKO
AG
091
XF+4,1
TRNS+5
LHHL
AH
0,1
XG+2 , 1
CLA
TRNS+6
LHML
AK
XH
TRNS+7
LHKL
AM
0,1
XK+8,1
TRNS+8
ISET,1
SETTR+3
TRABD
BA
TRABB
STO
BD1
AM
LXD
AN
XEC
CLA
STO
ROTATION AXIQ IS C, Bs
RuTATION AXIb IS B
OR A
q28
AO
BA
BB
CLA
STO
TRA
CLA
STO
CLA
STO
CLA
STO
TRA
NOP
CLA
CAS
TRA
TRA
ADD
TZE
TRA
SSP
BB1
SC
Bcl
BD
BD1
BB2
bC2
BD2
CA
STO
TRA
NOP
CLA
CAS
TRA
TRA
ADD
TZE
TRA
SSP
STO
TRA
NOP
CLA
CAS
TRA
TRA
ADD
TZE
TRA
SSP
STO
TRA
TRA
CLA
STO
TRA
CLA
STO
TRA
CLA
ST 0
TRA
R EM
CLA
TRAEX
BCl
CA
T RABC
BA
TRABB
BD1
TRAEX
i-<TATION AXI z
I
CA
MH
MH2
E RROR
BB2
ONE
*+2
*+2
1iCI-LMENT
MK
MK 2
E RR OR
BC2
ONE
*+2
MK
CA
IAC N LL ENT
L
IL2
ERROR
8D2
ONE
*+2
*+2
ML
CA
EX
MHl
MH
BB1
MK1
MK
MyL I
ML
B01
PLACE
ML
Ht
K,
ANG
L
IN
I
iul)rTUR
A
DAl
DA2
DA3
DA4
DPi
DB2
DC1
DC2
DD1
DD2
DD3
DD4
SSP
ARS
A)M
ALS
ADM
PAI
TRA
REM
OFT
TRA
CLA
TRA
OFT
TRA
CLA
TRA
OFT
TRA
CLA
TRA
OFT
TRA
CLA
TRA
OFT
TRA
CLA
TRA
OFT
TRA
CLA
TRA
OFT
TRA
CLA
TRA
OFT
TRA
CLA
TRA
OFT
TRA
CLA
TRA
OFT
TRA
CLA
TRA
OFT
TRA
CLA
ADD
TRA
OFT
TRA
18
MH
9
MK
T RNS
EXTINCT ION RULL NOUTINL
0 L
YA
1,2
ML
TEST2
YA
1,2
ML
TEST3
YA
192
ML
TEST4
YA
192
ML
TLST6
OKO
Yb
1,2
MK
TE.ST2
YB
1,2
MK
TEST4
HOG
YC
1 ,2
MH
TEST2
YC
1,2
MH
TEST4
HOL
YD
192
ML
TEST2
YDL
1,2
MH
TEST2
YD
1,2
ML
MH
TEST2
YD
1,2
230
DEl
DE2
DE3
DE4
DF1
DF2
DF3
DF4
ML
18
CLA
ARS
LBT
TRA
TRA
CLA
ARS
LBT
TRA
TRA
CLA
ADD
TRA
OFT
TRA
CLA
TRA
OFT
TRA
CLA
TRA
OFT
TRA
CLA
ADD
TRA
OFT
TRA
CLA
ARS
LBT
TRA
TRA
CLA
ARS
LBT
TRA
TRA
CLA
ADD
TRA
OFT
TRA
CLA
TRA
OFT
TRA
CLA
TRA
OFT
TRA
CLA
ADD
TRA
OFT
*+2
BA
MH
18
*+2
BA
ML
MH
TEST4
YE
1o2
MK
TEST2
,
OKL
YE
192
ML
TEST2
YE
1,2
MK
ML
TEST2
YE
1,2
MK
18
*+2
BA
MH
18
*+2
BA
MK
ML
TEST4
YF
192
MH
TEST2
YF
1,2
MK
TEST2
YF
1,2
MH
MK
TEST2
YF
HKO
?31
DG1
DG2
DH1
DK1
DK2
DK3
DK4
Dr,5
DK6
TRA
CLA
ARS
LBT
TRA
TRA
CLA
ARS
LBT
TRA
TRA
CLA
ADD
TRA
CLA
SUB
TZE
TRA
CLA
TRA
CLA
SUB
TZE
TRA
CLA
ADD
ADD
TRA
CLA
ADD
TZE
TRA
CLA
TRA
CLA
ADD
TRA
CLA
ADD
TRA
CLA
ADD
TRA
OFT
TRA
TRA
ONT
TRA
TRA
CLA
ADD
ADD
TRA
CLA
CHS
1.2
MH
18
*+2
BA
MK
18
*+2
BA
MH
MK
TEST4
MH
MK
*+2
192
ML
TEST2
MH
MK
*+2
192
MH
MH
ML
TEST4
MH
MK
*+2
192
ML
TEST2
MH
MK
TEST2
MK
ML
TEST2
ML
MH
TEST2
YK
*+2
EA
YK
BA
EA
MH
MK
ML
TEST2
MH
HHL
H(-H)L
HKL
?7~2
DK7
DK8
EA
TRNS
TEST2
TEST3
TEST4
TEST6
THREE
FOUR
SIX
XA
XB
ADD
ADD
TRA
CLA
SUB
ADD
TRA
CLA
SUB
TRA
LXD
LXA
TRA
REM
NOP
NOP
NOP
NOP
NOP
NOP
NOP
NOP
NOP
TRA
REM
S$P
ARS
LBT
TRA
TRA
SSP
LRS
DVP
TZE
TRA
SSP
LRS
DVP
TZE
TRA
SSP
LRS
DVP
TZF
TR A
OCT
OCT
OCT
REM
TSX
TSX
TSX
TSX
TSX
TSX
MK
ML
TEST3
MH
MK
ML
TEST3
MH
MK
TEST3
SAVE 1
SAVE,2
14
SECTION
ASSEIBLINO
0OL
OKO
HOO
HOL
0KL
HKO
HHL
H (-H) L
HKL
TO EXIT
EA
ROUTINES FUR TESTING
FOR
TRAraFER IN"STrUCTIONS
INDICES
FIT EXTi4CTIUN
EXTINCTION
RULL SELECTIUN
nWHETHL
EA
36
THREE
FOUR
EA
BA
36
SIX
EA
BA
000000000003
0000000Q0004
000000000006
TRANSFER INSTRUCTIUNS
00L
DA4,2
DA3,2
DA2 ,2
DAl ,2
OKO
DB,2 '2
DB1 ,2
rUN
RULE
-
~WE
~
2D33
XC
XD
XE
XF
XG
XH
XK
TRABA
TRABB
TRABC
TRABD
TRAEA
TRAEX
SLTTR
YA
YB
YC
YD
YE
YF
YK
SAVE
ONE
EX
EPROR
MH
MK
ML
MHl
MH2
MK1
MK2
TSX
TSX
TSX
TSX
TSX
TSX
TSX
TSX
TSX
TSX
TSX
TSX
TSX
TSX
TSX
TSX
TSX
TSX
TSX
TS X
TSX
TSX
TSX
TSX
TSX
TRA
TRA
TRA
TRA
TRA
TRA
TRA
TRA
TRA
RCM
OCT
OCT
OCT
OCT
OCT
OCT
OCT
PZE
OCT
CALL
PSE
TRA
C OMMON
COMMON
COMMON
COMMON
COMMON
COMMON
COMMON
COMMON
HOO
DC2,2
DC1 ,2
HOL
DD4,2
DD3 , 2
DD2,2
DD1,2
DE4,2
OKL
DE3 , 2
DE 2,2
DE1t2
HKO
DF4,2
DF3 , 2
DF2 , 2
DFl ,2
HHL
DG2,2
DG1 ,2
H(-H)L
DHi,2
DK8,2
HKL
DK7 ,2
DK6 ,2
DK5 ,2
DK4,2
DK3 , 2
DK2 , 2
DK1 2
BA
BB
BC
BD
EA
EX
ROTATI ON AXI)
AO
I
A
AN
ROTATI ON AXIo IS B
RO TAT ION AXIS IS C
CA
MASKS IOR IDENTIFYIN G SPECIAL REFLECTIONS
0 0L
777777000000
0 KO
777000777000
H 00
000777777000
H OL
000777000000
777000000000
0 KL
000000777000
H KO
H KL(ODD OR EVEN)
001001001000
000001000000
TERM
141
EX
7
1
1
3
1
1
1
1
TURN ON
SENSE LIGHT
1
ML1
ML2
I SET
LHKL
LHKO
LOKL
LHOL
LHHL
LHML
LHOO
LOKO
LOOL
COMMON
COMMON
COMMON
COMMON
COMMON
COMMON
COMMON
COMMON
COMMON
COMMON
COMMON
COMMON
END
235
Appendix IV
Observed and calculated structure factors
for wollastonite and pectolite
Wollastonite structure factors
The FCAL t s for wollastonite were computed using the
coordinates and isotropic temperature factors given in Chapter 1.
Separate scale factors assigned to each reciprocal lattice level
were varied in the last refinement cycles.
The scale factors used
for Appendix IV FCAL s were 2. 2127, 2.4003, 2. 1966,
2. 3820, 2. 2853,
2.3164, 2.3359, 2.4079, and 2.4977, for levels zero through eight,
respectively.
The contribution to the structure factors due to the
imaginary component of the anomalous scattering of Ca is listed
under BOBS and BCAL for both wollastonite and pectolLt4.
H
-0
1
2
3
4
5
6
7
8
9
1
K
-0
-0
-0
-0
-o
-0
-0
-0
-0
-0
-0
-0
-4
4
-2
2
3
3
-5
5
7
-0
7
-0
6
6
-8
8
9
9
-4
4
-2
2
-3
3
-5
5
7
7
6
-8
8
9
-0
1
-4
4
-2
3
-0
-1
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-o-0
-1
-1
-1
_2
-2
-1
-1
-1
7:-1
-1
-1
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-2
-0
-0
-0
-0F
-0
2
-3
L
-1
-0
-0
-0
-0
-0
-0
-0
FOBS
20.03
37.96
59.94
52.44
52.65
-3
-3
57.36
AOBS
-18.78
-37.92
-59.92
-52.07
52.49
61.90
61.94
70.31
35.53
33.99
35.28
27.31
28.13
25.62
16.04
15.75
46.40
16.22
24.95
26.89
23.32
14.69
10.46
48.90
14.86
-26.34
-28*09
-25.61
13.35
15.33
-45.86
-15.98
26.68
24o84
- 26.68
24.43
40.00
5.78
57.97
28.29
22.36
16.07
23.78
60.82
17.20
13.78
1.79
9.83
82.64
82.40
25.31
10.25
64.77
46.71
61.62
46.34
94.98
85.29
30.02
42.50
5.66
62.61
27.57
20.28
15.15
22.29
68.47
15.30
13.23
2.84
8.95
94.51
100.67
25.37
11.76
64.52
52.81
65.46
50.73
114.54
102.69
39.25
1.90
-23.89
39.99
0.00
57.91
28.13
21.72
16.06
-23.63
-60.47
-16.07
13.77
-1.31
6.47
-82.58
-82.29
-25.31
10.14
64.34
46.35
61.61
46.30
-94.78
-85.05
-25.02
44.21
58.49
90.89
28.86
23.26
29.43
24.14
41.86
63.59
15.51
80.35
1323
103.00
27.51
24.88
29.46
29.54
20.45
8.24
2.97
25.02
44.57
58.80
15.81
91.71
16.54
-3
FCAL
6.67
22*19
64.73
60.14
-0
-3
29.65
23.6
-0
-3
8.70
15*59
3 9.*45
2.37
-15.46
-91.65
-16008
90.67
ACAL
-6.25
-22.17
-64.71
-59.72
57.19
70.26
33.55
-24.07
-26.85
-23.31
12.22
10.18
-48.32
-14.64
-24.84
-29.35
42.49
0.00
62.53
27.41
19.70
15.14
-22.15
-68.08
-14.30
13.23
-2.09
5.89
BOBS
-6.98
-1.87
-1.61
-6.21
4.08
2.30
4.22
-7.19
-1.41
-0.79
8.90
-3.62
-7.12
2.73
0.16
-5.12
BCAL
-2.32
-1.09
-1.74
-7.12
4.45
2.62
4.01
-6.57
-1.34
-0.72
8.15
-2.41
-7.50
2.50
0.15
100.54
-25.37
11.64
64.09
52.40
7.40
-3.00
-4.17
-0.05
1.44
7.44
5.80
-6.30
-0.93
5.66
3.05
2.93
4.80
-0.57
2*50
-7.36
-5.45
0.20
-1.93
6.74
-3.43
-5.09
-0.05
1.65
7.42
6o56
65.44
1.21
1.29
50.68
114 * 30
102.39
39.o19
1.51
1.99
-6.15
-6.43
2*17
-7.42
-7.74
1.79
1.14
-24.14
0.37
0.35
41.53
50.59
-94.45
63.25
-15.17
-80.30
-12.87
102.75
-0.87
5.78
2.83
3.01
5.29
-0.60
2.67
-6.54
-6.13
0.21
-1.21
6.04
-3#23
-3.27
-2.86
-3*84
-3*0/
6.24
-27.42
2.35
-22.13
26.92
-29.61
-23.68
29.04
-7.16
-'.2#8
1.65
22.4/
8.69
8.23
6.53
-3.29
-28.76
-29.50
19.46
5.25
7.07
2.24
-7.66
-4.3
/.25
1.65
6.28
0.29
0.28
-5
5
-7
7
-6
6
-8
-0
-0
-0
-0
-0
-0
-0
8
9 -0
-0 -0
-0
-1
1 -o
-0
-4
4 -0
-0
-2
-0
2
-0
-3
-0
3
-0
-5
5 -0
-0
-7
7 -0
-6
-0
-0
6
8 -0
-0 -0
-0
-1
1 -0
-0
-4
-0
-0
-2
2 -0
-0
-3
3 -0
-0
-5
-0
-0
-7
7
-0
-6
8
-1
-4
4
-2
-2
-3
3
-5
5
7
-6
6
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-4
-4
-4
-4
-4
-4
-4
-4
-4
-4
-4
-4
-4
-4
-4
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-0
-0
-0
-0
-04
12.49
5.97
0.89
53.60
18.17
5.92
16.47
4.53
9.36
4.32
5.15
54.54
19.16
58.64
70.44
10o91
18.51
32.49
70.28
50.98
47.96
51.71
27.28
46.51
22.17
71.31
16.76
114.70
13.61
19.14
17.31
17*89
26.72
32.90
16.51
15.95
29.80
11.73
5.45
16.69
-5
-5
-6-5
-o-0 -65
-6
-0
-0 -6
-6
-0
-0
-0r -6
-0 -6
-0
-0 -6
-0
0.
49.51
18.69
5.78
18.65
5.55
11.08
3.97
4.64
58.84
21.57
55.78
64.00
12.18
20.79
33.08
66.81
48.01
46.74
46.06
27.06
47.07
23.06
62.52
18.96
98.99
15.59
21.82
18.78
17.92
27.28
34.17
18.11
17.96
31.34
-6
-6
-6
-6
-0.37
53.59
18.17
0.34
-14.96
4.06
-9.19
0.79
-4.63
-54.42
-19.06
70.27
-10.66
-18.26
-32.48
-70.10
-50.48
47.87
51.64
-26.61
-46.23
22.11
71.14
-16.61
114.47
-11.79
19.00
-16.52
-1 75)9
26.71
-33.57
-32.32
-6.38
-17.85
17.61
31.15
11.89
5.49
-16.27
15.64
29.62
11.17
5.01
-3.05
-16.49
-13.43
-2.55
-4.26
-4.82
-35.6t
12.73
-5e52
-5eb2
-0.56
-0.54
0.25
-0.48
58.63
0.
1.23
0.20
5.78
-7.80
2.46
2.07
-3.90
-2.04
-3.90
-2.26
0.94
4.41
2.55
-3.41
-1.08
0.81
1.33
0.20
5.91
-6.89
2.01
1.75
-0.
49.49
18.69
0.33
-16.94
4.98
-10.88
0.73
-4.17
-58.71
-21.45
55.77
63.85
-11.91
-20.51.
-33.06
-66.64
-47.54
46.66
45.99
-26.39
-46.79
23.00
62.37
-18.79
98.80
-13.51
21.67'
-17.93
-1763
27.28
-4*67
-6.70
2.85
2.42
-5.95
-5.13
1.63
4.29
-2.56
6.17
7.79
-2.57
-5.60
-4*25
-2.26
-3.62
-2*00
0.98
4.85
2.28
-3.04
-1.06
-4.91
-7.11
2.93
2.72
-6.00
-5.07
1.57
4.90
-2.27
7*15
6.80
-2.26
-5.16
3623
3.22
-0.40
-0.39
-6T14
-2.78
3.12
3.26
3.51
3.43
3.81
2.36
2.94
3.58
2.15
5.17
35o49
13.03
29.38
2.42
5.86
36.10
12.74
2z6.08
0.53
0.21
-2.22
61.92
69.23
-61./6
-69.06
-4o33
27.18
28.03
6.42
63.06
31.28
1.20
48.29
51.93
51.23
16.83
-27.09
-27.93
6.07
62.77
31.05
-0.50
48.17
-51.68
-2.27
-4o84
-2o34
2*58
2o07
5.67
6.05
3.86
3o75
4.15
3.39
-4.46
1.08
3.47
-5.06
-50.90
-5*45
-5*82
7.99
59.14
32.21
4.58
47.22
45.79
47.96
18.12
12.08
11.99
10.18
-3-506
13.02
U.28
0.97
7.56
58.86
31.98
-1.92
47.10
-45.58
-47.65
-18.10
-11091
-11.82
-16.81
-90.95
0.81
1.099
-10o04
-2.01
3933
0.75
-1.71
238
-0
-0
-1
-2
-3
-4
1
2
3
4
5
6
-1
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
1
-0
-0
2
3
-0
4
-2
-0
-9
-8
-7
-6
-5-4
-3
-2
-1
-0
1
2
3
4
5
6
7
8
9
1
-1
-1
4
-4
-4
2
-0
-0
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
3
-3
-3
53
5
-8
-8
-8
-8
-0
-0
-0
-0
-0
-0
-0
-0o
-o
-0o
-0
-0
-0
-0
-0
-0
-0
-0
1
-1
1
-1
54.84
20.09
10.84
4.27
6.68
5.32
31.49
5.58
0.
9.54
0.
28.44
12.19
3/(.05
16.29
37.47
2.81
32.23
22.68
3.94
17.60
31.02
6.59
1.14
18.17
6.37
34.59
13.75
0.
0.
28.66
23.94
9.08
9.18
4.27
20.98
21.58
7.08
18.31
59.49
18.32
10.40
3.97
7.09
4.31
32.65
4.00
0.60
8.86
2.88
29.10
10.87
-59.41
-18.01
9.65
3.81
5.03
-3.64
32.39
2.02
-0.32
-6.81
2.61
29.03
10.73
40.73
-16.00
-2.96
-3.66
-4.03
-1.22
4.71
2.86
3.98
4.82
0.
-6.11
-0.
1.93
-1.96
2.91
3.21
16.01
-54.76
-19.75
10.06
4.09
4.75
-4.49
31.24
2.81
-0.
-7.33
0.
28.37
12.03
36.94
-16.28
-0.64
-0.63
43.53
-3/.29
-43.32
-3.62
-l57
-31.83
22.36
3.94
17.59
-31.01
6.59
-0.67
-2.33
-32.41
-5.08
-4.20
-1.00
-5.17
24.77
3.49
16.34
-30.84
3.80
0.19
0.62
-0.88
-0.04
-0.26
0.64
-0.13
-0.09
-0.23
0.
0.
-0.12
0.25
-0.13
0.18
-0.67
0.37
-0.05
0.73
-0.61
4.09
1.20
32.82
25.12
3.50
16.35
30.86
5.70
0.49
18.31
6.10
40.63
12.61
1.77
0.10
30.20
26.63
8.41
8.87
4.09
20.54
22.17
7.44
16.31
10 /3
37.90
T9*06
4.58
21.91
36.14
40*86
5.70
-4.08
-4.57
24.00
21090
7.61
-6.72
23.98
-7.57
-754
-6.56
0.33
0.28
41.60
42.67
-0.37
-0.38
-0.81
-0.77
U.41
0.43
0.32
-U./U
0*28
-34.59
6.75
-1
-1
41.60
42.67
-1
-1-1
19.39
18.63
-19.38
22.00
23o93
-22.69
5.10
-18.61
-23o92
-5.09
-289i
27.83
-3191
-11.83
-1
-1
7.55
26.68
5.65
27.83
3.91
-4.90
-26.48
26.68
-1.65
12.53
11.83
-12.53
28.92
-1
~1
-1,
4.91
4.21
0.17
0.58
-0.88
-0.04
-0.11
0.65
-0.13
13.75
0.
-0.
28.66
-23.94
-9.08
9.18
-4.21
20.98
-21.58
7.04
-18.30
11.86
36.14
18.16
6.37
-1
-1
-3.21
-3.33
-3.87
-1.14
4.99
2.32
4.12
3.46
0.50
-5.67
-1.21
1.98
-1.75
-0.48
18.30
6.09
-40.63
12.61
1.77
-0.01
30.20
-26.63
-8.41
8.87
-4.04
20.54
-22.17
7.40
-16.30
100 0
37.90
-loll
-1
-1
-2
-2
-6
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-/
-8
-8
-8
-0.49
-0.28
0.780
-0.64
0.31
-U.D4
0.27
-U.UU
-0.00
-0.10
-0.21
0.06
0.10
-0.12
0.27
-0.12
0*17
-0.64
0.36
-0.05
0.77
-0.54
0./0
-0.52
-0.31
0.88-
-0.72
-0.00
-0.000
239
5
-5
7
-7
-7
6
6
-6
-6
8
8
-8
-8
9
9
-9
-0
-0
1
-1
-1
4
4
-4
-4
2
2
-2
-2
3
3
-3
-3
5
5
-5
-5
7
7
-7
6
6
-6
8
8
-8
-8
9
9
-97
-0
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1-
-1
-1
-1
-1
-1
-1-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-l
-1
-1
-1
1
-1
1
-1I
1
-1
1
-1
1
-1
1
-1
1
-1
1
-l1
1
1
-1
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
2
-2
2
-2
2
-l
-1
-2
-l1
2
-1
-2
-l1
-1
2
-2
2
-2
6.63
0.
0.
27.85
7.38
6.19
22.16
10.76
1.50
4.75
21.79
13.46
10.88
7.15
10.91
2.15
6.58
0.
4.52
9.87
10.19
19.43
44.93
22.06
7.76
13.51
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47.87
29.71
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16.42
55.03
48.07
48.37
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34.79
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47.70
49.28
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250
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252
7
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78.99
146.84
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78.78
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63.93
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6.41
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-1.41
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3.15
-6.57
-0.18
7.06
-2.55
6.92
-2.24
-2.14
6.79
3.62
-1.18
-0.99
-0.33
-7.05
-6.93
-0.08
3.46
3.18
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3.28
1.11
0.89
3.14
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4.71
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2.11
4.33
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5.12
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0.58
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2.95
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7037
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6.58
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6.61
4.13
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-5.75
-0.08
4.01
2.89
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-6.82
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2o87
1.11
0.89
2.90
-4073
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6.86
4.78
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1.80
4.63
6.75
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4.81
2.06
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0.52
5.80
5.95
254
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20.57
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8.62
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15.61
22.14
2.95
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36.30
36.31
11.09
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37.09
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255
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39.34
6.58
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29.49
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34.30
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22.47
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22.64
28.85
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2.81
16.62
30.23
3.18
48.67
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30.43
31.55
28.24
6.56
17.19
25.36
26.99
20.25
10.70
10.32
39.14
8.86
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12.17
31.66
6.73
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12.63
36.08
18.02
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32.49
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26.71
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14.86
33.24
14.61
1.69
3.87
39.33
6.53
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2.06
41.87
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29.83
26.78
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27.03
23.45
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10.97
12.40
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10.30
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19.58
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30.01
6.86
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13.65
14.78
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18.11
3.50
-22.78
17.06
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15.88
-7.86
26.00
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9.00
12.96
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16.20
2.01
4.35
39.69
2.79
-16.62
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3.18
48.66
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31.55
28.24
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26.98
20.25
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10.28
12.12
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8.81
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18.55
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31.65
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12.62
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18.01
3.91
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3.77
1.06
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0.39
0.69
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0.07
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0.40
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0.33
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0.38
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0.97
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0.56
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1.29
256
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32.62
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33.22
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34.63
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43.68
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32.57
28.08
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30.41
34.62
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32.37
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28.04
34.43
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32.14
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11.71
15.75
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3.86
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12.70
16.72
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18.44
6.83
5.75
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3.42
4.66
0.89
6.13
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5.67
1.83
23.97
16.94
4.15
17.05
3.17
13.04
2.92
4.18
2.64
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24.69
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3.77
11.85
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27.41
31.60
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16.89
32.34
15.77
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16.94
3.86
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6.45
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33.09
18.10
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27.74
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6.40
16.77
21.23
16.27
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15.69
2.87
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34.18
16.37
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27.39
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5.76
16.85
32.28
0.37
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1.17
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1.58
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1.19
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0.39
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0.53
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1.90
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0.71
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1.61
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0.39
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1.26
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1.27
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1.02
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0.06
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1.12
0.60
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0.39
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0.47
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0.36
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1.87
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0.82
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0.70
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1.51
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1.43
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0.53
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1.56
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1.82
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2.32
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U.96
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1.42
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1.42
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0.95
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0.49
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1.65
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0.94
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1.47
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1.93
0.39
257
3
3
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4
4
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2
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23.72
22*17
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18.84
25.43
16.15
28.81
17.53
17.37
19.27
21.01
18.86
19.99
7.00
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48.69
6.33
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17.02
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19.18
20.94
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5.72
20.03
15.77
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16.06
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18.78
19.92
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5.49
19.45
16.93
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2.00
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1.71
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0.95
1.78
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-1.82
1.00
-1.86
0.94
1 o75
-1.67
1.60
-1.71
0096
-1.51
-1.06
1.64
1.65
-1.87
1.25
-1.45
-1.02
1.76
1.61
-0
-1.47 6+
1.14
.71
7
-5
4
15.88
16.23
-15.78
-16.12
1.80
1.84
6
6
8
-0
-0
1
1
-1
-1
-5
-5
-5
-5
-5
-5
-5
-5
-5
-4
4
-4
-4
4
-5
5
-5
5
8.12
13.37
10.98
46.37
5.87
7.81
10.30
2.60
19.13
2.10
13.26
10.75
45.65
3.68
6.30
8.60
2.61
19.86
-6.43
-13.27
-10.93
46.32
-5.41
7.67
-10.09
-0.58
-19.05
-1.66
-13.16
-10.70
45.59
-3.39
6.19
-8.42
-0.58
-19.77
4.96
-1.70
1.08
2.32
-2.26
1.45
-2.09
2.53
-1.82
1.28
-1.69
1.06
2.29
-1.42
1.17
-1.75
2.54
-1.88
-5
-5
-2.10
1.49
-2.08
-1.31
-2*04
2.02
1.94
2.01
-0.
0.
-1.10
1.39
2.01
-2.00
1.88
-1.21
1.37
1.*58
-1.78
1.39
1.53
-2.01
-2.00
0.
-2.26
-1 .96
1.90
-2 .14
2.11
3
-3
-3
4
4
-4
2
2
-2
-2
5
5
7
-5
-5
-5
-5
-5
-5
-5
-5
-5
2
5
-5
5
-5
5
5
-5
5
-5
-5
-5
-5
-5
6
-5
6
-0
-0
-5
-5
5
-5
5
-5
5
1
1
-1
-1
-5
-5
-5
-5
-5
-6
6
-6
6
.-
t
0
2
.
67J
10
:)U
21.40
14.12
20.81
15.54
21.29
14.04
8.49
28.83
7.77
29.62
22.63
-8.23
23.56
25.17
0.
0.
34.37
28.68
16.04
34.13
25.21
2 * 12
1.96
37.92
28.15
15053
38054
0.
-0.
-34.35
28.64
15.96
0.69
-0.55
-37.90
28012
15.45
-38.49
22o96
33.08
-22.
-22.87
34.53
12.45
0.
16.34
12019
2.07
31.05
30.85
25.09
15*45
-7.53
-29.59
22.54
25o 13
-34.08
33.19
-12.28
-0.
16.18
-7.16
33.22
7*52
22.35
-28.80
23.48
1 * "t'1
20.71
15.47
6.89
25
30.98
35.74
-30.77
2.55
1.63
20.92
5.71
-2.35
34.50
-12.03
-0.83
15.32
-6.55
33.00
-35065
2.31
-2.*03
2.07
-2.18
1 .64
-1.90
-1034
1.45
-2.09
2.20
-20 52
3
3
-3
-5
-5
-5
-6
6
6
3.22
0.
19.75
6.10
4
4
-5
-5
-6
6
8.49
30.33
8.44
32.72
2'
-5
-6
32.67
34.14
32.57
34.03
2.53
2.64
2
-5
11.09
10.31
-10.88
-10.11
-2.17
-2.01
6
2.19
1.74
0.
19.65
5.60
-1.86
0.08
20.81
5.24
-0.
2.01
2.43
-1.63
2.13
2.27
8.32
-30.27
8.27
-32.65
1.68
-1.93
1.67
-2.09
258
-2
-2
5
6
-0
-0
1
-1
3
2
2
-0
-5
-4
-3
-2
-1
-0
1
2
3
4
5
6
7
8
1
-13
3
3
-3
-3
4
-4
-4
2
2
-2
-2
5
5
-5
-5
7
7
6
6
8
8
-0
-0
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
6
-6
-6
-6
6
-7
7
7
-7
-7
7
7
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
15.15
21.11
21.14
11.14
5.50
17.18
4.31
4.24
7.95
17.11
16.02
6.25
25.46
25.01
5 . 22
26.79
24.03
19.37
49.82
14.74
18.13
51.73
18.28
22.72
27.96
8.81
27.21
27.57
1
-1
-1
1
-1
-1
-1
1
-1
1
-l1
1
-l1
1
-l1
1
-1
-l1
1
-1
1
-l1
-l
8.46
61.26
13.27
28.40
8.81
0.
34.72
12.52
59.99
29.40
17.01
21.38
22.02
13.04
5.57
17.43
4.69
5.35
7*04
18.15
18.58
7.66
27.38
24. 74
6.19
24*73
23.76
19.84
63.86
17.31
20.32
65.64
18.70
24.07
27.19
5.64
26.13
25.81
8.27
73.33
13.17
31.41
8.08
-3*70
-3*84
-4684
-7.58
17.04
-15.94
-5.98
25.39
-24.94
-6*72
18.07/
-18.49
-7.35
27.30
-24.67
-3.17
24.20
23.66
19.65
-2,o67
26.22
23.92
19.18
-49.50
14.73
-18.12
51.40
-18.13
-22*63
-27.42
3.68
27.13
27.54
-4.94
-60.95
13.04
-28.22
8*78
29.43
28686
8.56
30.46
20.39
29.95
29.91
8.02
28.01
20.40
29.53
32.32
19.27
44.92
5.75
7.75
29.70
31.35
2.31
7.79
42.96
7*34
35 . 16
16.81
-21.29
-21.90
-12.91
5.41
-17.30
-3.40
72*74
32.48
12.50
9o76
44.23
6.57
9.31
31.98
32.50
0.
9.26
43.04
8.90
33.44
-17.05
-0.
-34.08
-12.28
9.68
2*03
11*28
18.78
14.97
-21.02
-21.03
-11.03
5*34
11.16
4.37
-30.42
20.33
29.66
-29.62
-18.74
-44.04
-6.47
-9.11
-31 *5
31.88
0.
9.13
42.86
-8.89
33.23
-63644
17.30
-20.32
65.23
-18.54
-23.97
-26.67
2.36
26.*05
25.78
-4.83
-72.96
12.94
-31.21
8.05
-0.99
-31
8
-12*27
72.36
28*88
9.66
4609
-27o98
20.34
29.24
-32.01
-19.23
-44.73
-5.66
-7.58
-29.11
30675
1.55
7.68
42o78
-7.33
34.95
2.34
-2.02
-2.25
1.55
1.31
-2.13
2.66
-1.80
2.62
-2*04
-2639
-2612
1 * Sb
-1.86
2.21
2.01
-1.85
-5.32
5.05
2.19
2 .74
-7.26
-0*53
0.28
-1.60
1.79
1.87
-4.49
'.47
2.22
2.68
-5.66
-0.45
0.25
5.81
-2.38
-2.04
- 5.47
-2
634
1.81
1.32
-2 o 16
2*89
-2.27
7637
-2 .43
-2. 16
2.12
1.27
-6.87
-5.32
5.13
2 04
1.18
-6.72
-6.15
-7.
2.46
-3.25
-0.75
0.
-6.66
-2.43
6.11
5*63
-1.63
7.36
-1.55
1.51
4.18
-4.10
-1.15
-4.10
-1.17
1.94
-b .31
6.32
-0.
1.52
3.93
0.53
3.69
2 .44
-3.60
-0.*68
1 .78
-6.23
-2 43
7.41
5.63
-1.41
6*89
-1.42
1.51
4.13
1.01
36
-4.43
-1.18
-4.17
-1 .03
1662
-b .86
6609
-1672
1
628
3692
0.44
3.88
259
1
-6
-2
1
-6
2
-1
-6
-2
-1
3
3
-3
-3
-6
-6
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
-2
2
2
-2
2
-2
2
-2
-2
4
4
-4
-4
2
2
-2
-2
5
5
-5
7
7
6
6
8
-0
-0
1
1
-1
-1
3
3
-3
-3
4
4
-4
-4
2
2
-2
-2
5
5
7
7
6
6
-0
-0
1
1
-1
-1
3
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
54.30
22.59
9.38
25.48
27.71
4.60
14.14
23.99
24.62
15.02
9.02
2.32
21.97
52.66
36.24
49.81
45.77
34.76
38.44
0.
8.56
23.50
17.51
36.00
53.21
19.33
8.44
25.15
28.42
4.08
13.00
21.63
53.95
21.88
9.26
25.31
-27.60
-3.90
-14.14
-23.97
26*19
-24.42
-25.99
11.20
8.94
2.80
18.83
49.98
37.46
55.00
-14.86
8.93
-11.09
8.86
2.80
-18.26
55.27
38.27
37.72
1.84
8.35
21.55
22.60
15.90
37.07
3.74
28.65
16.76
5.20
22.10
63.37
15.24
23.57
18.95
2.59
63.50
24.40
1.65
21.99
3.46
18.98
16.28
15./4
3
13.43
13.11
1146
12.42
3
-3
3
-3
3
-3
3
-4
4
-4
4
-4
17.98
24.83
3.46
4.35
19.56
17.01
23.50
27o.39
25.85
13*38
2
3.55
29.65
-3
3
-3
3
-3
3
-3
3
-3
3
-3
3
19.66
5.54
22.95
57.89
16.51
25.86
18.54
3.26
54.69
25.27
2.11
24.77
3.48
-3
3
14.77
20.93
12.50
41.59
7.56
23.67
2.32
-21.31
-52.30
-35.60
-49.42
45.43
34.18
37.84
-0.
-8.48
23.48
17.50
-35.39
3.03
29.53
-18.67
-4.97
22.80
-57.54
-16.38
-25.38
-17.49
-2.98
54.36
-25.11
-1.83
23.74
2.55
20.64
16.12
13.42
-11.46
-17.79
52.87
18.73
8.32
24.98
-28.31
-3.45
-12.99
-21.62
-49.64
-36.80
-54658
54.86
37.62
37.13
-1.83
-8.27
22.58
15.90
-36.44
3.19
28.55
-15.92
-4.66
21.96
-62.98
-15.12
-5.26
2.31
2.50
-6.97
-1.94
-23*13
-4.97
-4.53
-17.88
-2.37
63.11
-24.25
-6.14
1.33
5.99
-2.79
-1.04
7.07
-2.36
6.19
-6.27
1.06
6.96
-2.70
-0.82
6.27
-2.35
5.46
-1.44
21.08
2.54
18.18
1.5058
13.11
-12.41
26.92
25.39
23.83
22.69
-6.79
-6.17
2.46
2.60
-6.36
-2.11
18.47
-7.51
-6.59
1.94
3.90
18.75
6.51
2.94
-2.49
-2.16
0.29
-0.81
-3.27
-1.62
1.23
0.02
-4.57
-5.78
-7.01
-6.80
6.72
7.00
6.65
0.23
-1.15
1.02
0.05
2.48
3.75
13.78
11.46
452/
2.98
-2.43
-2.44
0.31
-0.90
-3.08
-2.18
1.24
0.01
-5.33
-6.09
-6.79
-6.16
5.57
6.36
6.77
0.
-1.18
1.06
0.06
2.56
-16.83
-22.59
3.57
3.81
17.70
14.66
-20.70
-12.49
41.4
6.01
4.81
1.38
1.85
-23.87
3.29
4.25
6.13
5.62
1.53
13.28
-13.63
-11.46
45*14
-6.47
22.84
2.29
2.21
-0.09
0.28
-0.09
0.30
-2.59
-2.45
-6.84
1.07
-0.91
6.46
5.10
1.76
-3.08
0.22
3.13
0.79
6./2
-6.47
1.16
-0.82
6.20
4.81
1.59
-2.02
0.20
3.40
0.68
6.77
260
3
-3
-3
4
4
-4
2
2
-2
-2
5
5
7
6
6
-0
-0
-1
-1
3
3
-3
4
4
2
2
-2
-2
5
6
-0
-0
1
1
-1
3
3
4
2
2
-0
-0
-3
-2
-1
-0
1
2
3
4
5
6
-6
-6
-6
-6
-6
-6
-6
-6
-6
4
-4
4
-4
4
4
-4
4
-4
-6
4
-6
-6
-6
-4
4
-4
-4
4
-4
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
4
-5
5
-5
5
-5
5
5
-6
-5
5
-5
5
-5
-6
5
-6
-6
-5
5
-5
-6
-6
-6
-6
-6
-6
-6
-6
-6
-6
-5
5
-6
6
6
-6
6
-6
-6
-6
-6
-6
-6
-6
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
6
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-6
-6
63.01
14.71
33.78
8.34
42.30
5.41
8*94
16.93
15.01
23.01
20.05
15.40
2.58
33.34
17.00
58.88
24.92
3.68
25.29
18.30
10.52
19.16
19.90
33.58
9.40
18.83
24.29
4.52
7.26
35.32
35.94
9.53
32.17
20.69
20.40
30.76
5.67
20.91
22.44
4.65
21.84
6.33
32.43
2.55
23.34
2.16
31.04
6.61
4.61
20.45
10.44
17.21
13.12
16.81
21.67
65.49
14.77
31.77
7.83
45.08
4.88
8.12
15.14
15.01
22.64
21.13
15.14
3.70
33.99
16 * 16
65.53
24.40
2.94
23.84
18.98
9.97
19.36
20.22
33.11
8.41
19.32
24.28
4.05
6.96
36.33
37.75
9.63
32.36
21.33
20.78
36.74
4.99
21.28
24.06
5.66
22.27
6.11
34.45
3.06
23.82
1.59
30.89
5..75
0.25
25.66
10.50
18.15
13.12
16.42
20.10
62.76
65*23
5.54
5*76
14.43
14.49
33.27
31*29
8.32
-42.20
-3.66
8.92
16.78
14.75
22.92
-19.98
-15.10
1.01
-32.85
-16.69
-58.66
-23.96
7.82
-44.98
-3.30
8.10
15.01
14.75
22.55
-21.05
-14.84
1.45
2.86
5.86
-0.50
-2.89
-3.98
-0.62
2.19
2.83
2.04
-1.69
-3.04
2.37
2.87
5.51
-0.47
3.56
-24.45
-17.49
10.10
-19.12
19.85
-33.31
-8.94
17.99
23.49
-4.34
-6.98
35.06
-35.66
9.40
31.91
-20.64
20.35
30.44
-5.16
20.69
-22.25
-4.27
-33.50
-15*86
-65.29
-5.65
-23.46
2.84
-23.05
-18.14
9.57
-19.32
20 * 16
-32.84
-8.00
18.46
23.49
-3.89
-6.86
0.95
-6*69
36.06
-37.45
9.50
32.10
-21.28
20.74
36.36
-3.25
-5.04
-6.44
-5.39
2.93
-1.21
1.45
-4.28
-2.89
5.55
6.18
-1.24
2.00
4.29
-4.48
-1.55
4.05
-1.51
1.40
4.39
2.36
-3*08
-3 .59
-0.56
1 *96
2.83
2*00
-1.78
-2*99
3*40
-5.76
-3.09
-5.61
-6.72
0.76
-6.07
-5.59
2 .78
-1.22
1.47
-4.22
-2.58
5*69
6 18
-1.*11
1.91
4.41
-4.70
-1 .57
4.07
-1 *56
1 *43
-3.34
5*24
2 * 07
3*08
-3. 15
-2*24
-3.40
-1.95
-1.88
2.40
-4.9
1.58
-1.08
1.090
23.59
1.57
30.89
5.69
-0.23
-25.65
-10.50
18.15
13.10
16.42
-20.10
3.25
-0.37
0.40
0.93
-1.79
-0.53
-0.23
0.21
0.62
-0.28
-0.03
3.32
-0.27
0.40
0.81
-0.10
-0.67
-0.23
0.22
0.62
-0.27
-0.02
-4.54
21*06
-23.85
-21o59
-5*19
-22.01
6.02
-32.8
5.81
-34.0/
1.99
23.12
2.12
31.03
6.54
-4.25
-20.44
-10.44
17.21
13.10
16.0bO
-21.67
3*02
-2.94
-1.85
261
7
1
1
-1
-1
3
3
-3
-3
4
4
2
2
-2
-2
5
5
7
7
6
6
-0
-0
1
1
-1
-1
3
3
-3
-3
4
4
2-
2
-2
-2
5
5
7
6
6
-0
-0
1
1
-1
-1
3
3
4
4
2
2
-2
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-0
-1
1
-1
1
-l
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-2
24.32
5.61
7.23
0.
24.71
0.
23.62
19.78
19.78
13.92
18.71
17.22
14.71
5.80
0.
23.09
3.68
12.11
14.22
16.35
19.20
28.41
0.
33.51
9.81
26.79
24.42
3.91
28.83
24.33
4.38
7.16
0.77
25.52
0.97
22.19
17.81
19.34
15.03
19.50
15.38
13.02
4.87
1.01
23.84
6.34
11.99
13.15
15.66
16.15
27.85
0.83
31.34
-2
5.99
4*64
-2
2
9.40
34.80
20.54
8.44
37.74
19.18
-2
-2
2
2
-2
2
-7
-2
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
2
-2
2
-2
2
-2
-2
2
-2
2
-3
3
-3
3
-3
3
-3
3
-3
3
-3
5.
19.05
9.49
13.58
0.
9.00
18.64
6.88
12.20
3.99
26.63
15.34
11.16
17.36
22.10
20.26
2.06
4.60
0.
9.48
21.06
19.17
9.43
27.71
25.13
3*18
27.13
4.60
17.02
9.49
13.57
0.75
9.08
18.30
7.89
11.53
2.14
26.48
15.25
10.36
16.67
21.76
21.36
2.54
3.19
1.73
8.54
20.08
20.79
-24.31
-5.56
-7.21
-0.
-24.71
-0.
23.62
-19.78
19.77
-13.92
-18.70
-17.19
14.67
5.60
-0.
23.08
-3.64
-12.10
14.21
16.35
-19.19
28.41
0.
33.48
9.80
-26.79
-24.38
3*88
-28*80
5.63
-9.39
-34.78
-20.52
-24.33
-4.33
-7.14
-0.05
-25.52
-0.96
22.19
-17.81
19.34
-15.03
-19.50
-15.36
12.98
4.70
-0.56
23.83
-6.27
-11.99
13.14
15.66
-16.14
27.85
0.79
31.31
9.43
-27.71
-25.09
3*16
-27.10
. 4.36
-8.43
-37.72
-19.17
-0.48
-0.79
0.55
-0.
0.65
0.
0.23
0.16
-0.53
0.07
-0.40
0.99
-1l.13
1.50
-0.
-0.60
0.58
0.17
0.59
0.01
-0.76
-0.32
0.
1.62
-0.31
0.06
-1.29
0.46
-1.37
2.05
-0.34
-1.10
0.74
5.22
4.59
19.01
-9.38
-13.58
0.
8.94
18.63
6.88
12.12
-3.61
26.62
-15.31
-11.09
-17.32
22.08
20.20
1.63
4.29
0.
9.45
-21.04
19.17
16.99
-9.38
-13.56
0.50
9.03
18.28
7.89
11.46
-1.94
26.46
-15.23
-10.28
-16.63
21.74
21.30
2.02
2.97
1.23
-1.42
0.13
0.
-0.97
0.71
-0.13
1.37
-1.70
0.95
-0.85
1.32
-1.16
0.77
-1.59
1.25
0*82
-0.
0.69
-0.90
0.41
8.52
-20.*06
20.79
-0.3
1.65
-0.48
-0.62
0.54
-0.77
0.67
0.17
0.22
0.14
-0.52
0.07
-0.42
0.88
-1.00
1.26
-0.83
-0.62
1.00
0.17
0.55
0.01
-0.64
-0.31
0*27
1.51
-0*30
0.07
-1 *33
0.37
-1 *29
1 . 59
-0.31
-1 . 19
0.69
-0).3
1.10
-1.42
0.13
0.56
-0.98
0.70
-0.15
1.29
-0.92
0.95
-0.84
1.22
-1.11
0.76
-1.67
1.54
1 .14
-1 *53
0.62
-0.85
0*45
262
-2
5
D
6
-0
-0
1
1
-1
-1
3
3
4
4
2
2
-2
5
-0
-0
1
3
2
2
-1
-0
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-8
-8
3
-3
3
-3
-3
3
-4
4
-4
4
-4
4
-4
4
-4
4
4
-4
-4
4
5
-5
-5
5
-0
-0
5.86
19.05
26.92
4.30
5.57
15.01
3.99
35.92
23.74
3.97
15.39
2.28
17.59
20.66
37.99
25.46
22.31
15.88
5.09
10.53
11.48
15.27
3.78
13.57
21.13
5.45
7.08
20.89
28.81
3.47
5.50
14.80
1.54
35.98
25.58
4.46
17.08
1.85
18.66
21.49
39.92
24.02
22.16
14.54
5.87
9.47
12.26
16.15
4.51
13.92
21.52
4.09
-5.84
-19.05
26.90
-4.07
-5.16
14.91
3.40
-35.86
23.63
3.87
15.31
1.56
17.58
20.58
-37.94
25.41
22.22
15.80
-4.96
-10.43
11.37
-15.24
-3.30
13.40
-20.33
5.09
-7.05
-20.88
28.78
-3.28
-5.10
14.70
1.31
-35.92
25.46
4.35
16.99
1.26
18.65
21.41
-39.86
23.97
22.08
14.46
-5.72
-9.38
12.13
-16.12
-3.94
13.76
-20.70
3.82
-0.54
-0.42
1.06
1.38
2.09
-1.74
2.09
-1.99
2.30
-0.89
1.53
-1.66
-0.71
1.78
-1.94
1.57
1.92
1.57
-1.17
1.46
1.64
-0.96
1.84
-2.10
-5.76
-1.95
-0.65
-0.46
1.14
1.12
2.06
-1.72
0.81
-2.00
2.47
-1.00
1.70
-1.35
-0.75
1.85
-2.04
1.48
1.91
1.44
-1.34
1.31
1.75
-1.01
2.20
-2.15
-5.86
-1.46
1
-8
-
13
15
-13
-15;
-1
-1
2
-8
-0
50.05
3
-8
-n
11.5
4
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-0
6.01
23.24
36.40
22.10
5
1
1
-1
-1
3
3
4
4
-0
-1
1
-1
1
-l1
~1
-l1
1
-1
8L
C
4.74
20.34
20.01
35.66
18.29
12.77
10.02
60
64.88
14.08
5.68
23.81
34.04
22.28
3.89
21*13
21.37
34.43
19.37
.
45
49.75
-11.53
5.86
-22.46
35.65
-22.01
-4.64
19.78
-19.94
34.95
-17.55
12.77
5
64.50
-14.06
5.54
-23.01
33.34
-22.19
-3.80
20.55
-21.29
33.75
-18.58
12.90
-8.30
-10.07
17.06
.
0)1
5.44
-0.65
-1.35
-5.97
7.36
-1.99
-0.97
4.72
-1.73
7.04
.
17
7.05
-0.80
-1.28
-6*12
6.88
-2*00
-0.80
4.91
-1*84
-. 5.16
6 * 80
-5.47
0.24
-2.16
-1.83
-2.65/
-2.a18
4.54
-0.65
-0.14
-5.19
-3.20
-4.09
4.63
-0 60
-8
-8
-8
-8
-2
2
-2
2
25.80
16.99
17.60
12.90
8.50
10.30
17.68
3.76
10.03
17.52
20.11
36.93
39.27
21.91
26.91
26.92
13.88
14.67
-u
-8
-2
26.80
274.7
(4
206.10
20 sUi
61
6
-0
-!8
26.57
26.56
25.97
25.96
5.63
5.63
2
2
5
5
-0
-0
1
1
3
4
4
2
2
12.61
-1
-1
1
-21
2
-2
2
17.31
4.06
10.19
17.67
23.33
36.69
38.12
24.50
24.35
-9.79
-12.32
16.70
-4.01
10.19
-16.89
-23.10
-36.46
-37.87
-24.26
23.85
25.12
-16.77
-17.37
-3.71
10.*03
-16*75
-19.92
-36.70
-39.01
-21.69
26*36
26.21
-13.69
-14.47
-4.36
-3.45
4.90
5.89
-2.74
-2.85
0*24
-0
14
-5.15
-2.76
-4.11
-4.*49
-3 . 09
5.42
6 . 15
-2.24
-2 37
26~3
S-8
3
-8
2
-8
2
-8
3
-3
-3
3
37.38
35.20
22.97
21.11
39.76
38.98
22.60
d0.46
36.95
34.79
-22.73
-20.90
39.32
38.54
-22.36
-20.26
L.65
5.30
-3.33
-2.95
6.01
5.87
-3.28
-2.86
264
Pectolite structure factors
The FCAL's for pectolLte were computed in the same way
as for wollastonite except that the coordinates and temperature
factors of Chapter IV were used together with a single overall
scale factor.
265
0
1
1
2
2
0
1
2
1
2
3
3
2
3
0
1
3
1
2
3
4
2
4
4
3
4
0
1
1
3
2
4
4
5
5
2
3
5
5
4
1
0
3
5
4
2
1
5
6
6
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
L
0
1
-1
1
0
-1
2
-2
1
2
-2
-1
2
1
3
-3
3
-3
2
0
-2
3
-1
1
-3
-2
4
-4
3
-4
2
-3
-1
4
-4
1
FOBS
12.43
16.49
22.45
7.11
40.57
5.21
44.90
60.32
58.74
72.67
55.67
62.72
13.00
16.63
25.01
40.91
71.65
51.37
7.88
34.35
42.69
31.79
23.41
42.23
50.17
4.66
25.53
73.96
27.05
51.21
52.93
51.45
12.23
60.12
40.44
41.36
19.09
19.11
87.12
49.69
3
-03
5
4
2
-4
-5
-3
-1
0
5
-2
41.18
FCAL
11.46
16.44
20.74
5.58
43.46
5.27
49.64
81.00
71.20
98.65
69.32
76.68
9.42
17.28
23.09
39.43
95.14
54.13
8.08
33.67
41.66
28.67
20.39
41.26
50.22
4.50
24.15
84.89
24.82
51.76
55.08
52.71
11.30
65.31
39.89
40.85
18.67
19.22
104.71
49.28
40.51
4.641
AOBS
11.90
16.11
22.40
-0.47
-40.54
-0.96
-44.89
-60.20
58.68
72.62
-55.53
-62.58
11.32
15.77
24.93
-40.59
-71.56
51.37
-7.82
-34.34
-42.59
-31.40
-23.01
-42.22
-49.90
2.39
25.18
-73.84
27.02
-50.99
52.68
51.32
-12.03
59.93
40.23
41.36
-19.07
18.36
-87.05
ACAL
10.98
16.06
20.69
-0.37
-43.42
-0.97
-49.63
-80.84
71.13
98.58
-69.15
-76.51
8.20
16.39
23.02
-39.12
-95.01
54*13
-8.02
-33.66
-41.57
-28.32
-20.04
-41.26
-49.95
2.30
23.82
-84.75
24.80
-51.55
54.82
52.58
-11.12
65,&10
39.68
40.85
-18.66
BOBS
3.58
3.50
-1.57
7.09
-1.56
-5.12
-1.11
-3.79
2.55
2.68
-3.84
-4.26
-6.40
5.27
-2.02
-5.13
-3.66
-0.55
-0.96
-0.82
2.91
-4.93
4.29
-0.36
-5.18
4.01
4.23
-4.26
-1.11
-4.70
5.12
3.59
-2.19
4.89
4.15
-0.12
-0.75
BCAL
3 *31
3*49
-1.45
5 .57
-1.67
-5*18
-1 *22
-5.09
3.09
3.63
-4.78
-5.21
-4.64
5.48
-1.86
-4.94
-4.86
-0.58
-0.98
-0.81
2*84
-4.45
3.73
-0.35
-5*18
3.86
4.00
-4.89
-1.02
-4.75
5.33
3.68
-2.03
5.31
4*10
-0*12
-0*74
18.46
5.30
5.33
104.63
-3.51
-4.22
49.41
49.01
').*2 /
41.10
40.43
2Z/4
-24.49
2.61
5.22
2 *57
-1.30
3.45
-1 *22
is./0
26.14
68.66
67.08
44.42
17.50
24.92
60.44
68.48
12.18
52.3
24.52
78.32
73.94
44.61
15.46
68.58
78.23
3.32
-66.92
-44.34
16.44
-73.76
-4.62
-44.o53
2.61
23.53
24.38
65.43
72.25
12.34
5.50
60.40
23.03
65*39
11.47
52.24
72.07
11.63
5.s43
5.58
5.76
3.48
3*58
7.59
7.79
/.28
/.48
-26.11
68.31
14.52
6.00
-5.15
2.24
4.82
4.08
2.64
-4.37
2.14
-5*10
2.63
5*30
-4.87
2.42
5.08
4*14
2.80
-4.50
2.19
266
5
5
4
3
6
6
7
5
7
3
5
6
7
7
3
1
0
6
1
5
8
8
7
3
4
8
8
2
8
4
8
7
6
6
3
7
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
4
3
-4
-5
5
-3
2
-6
6
-6
6
4
0
-6
-5
-4
6
3
-2
5
1
-6
-3
2
6
-7
7
-5
4
5
-7
-4
7
-6
0
3
-7
6
-2
1
7
-3
-7
2
5
5
7
4
6
-4
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21.75
23.89
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5.15
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0.81
15.01
4.59
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62.88
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34.02
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0.88
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3.74
2.48
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3.73
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14.02
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2886
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4.99
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4.20
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287
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3.76
10.17
61.96
3.80
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54.61
56./3
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5.53
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13.42
17.88
19.15
27.38
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18.62
26.71
14.20
17.89
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13.00
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21.63
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8.50
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34.86
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29.67
31.55
32.42
44.62
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51.32
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15.69
15.84
3.54
14.41
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36.74
28.95
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32.97
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3.59
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25.14
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4
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8.80
2.93
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26.79
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36.32
10.56
37.46
31.81
28.54
28.79
31.55
33.65
47.33
48.50
32.99
57.99
9.15
59.42
14.77
13.96
1.62
12.26
12.34
37.24
27.54
12.84
31.56
26.90
6.08
2
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6
6
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7.66
9.47
4.88
31.20
27.82
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35.09
13.43
34.96
31.17
30.23
29.87
31.92
32.76
44.84
45.20
31.74
51.52
11.93
55.94
16.26
15.98
3.61
15.33
13.48
37.06
29.35
16.62
33.02
27.54
7.59
32.85
28.28
3.65
8.39
58.34
2.71
54.48
23.39
-14.85
17.86
18.73
24.51
4.64
16.76
-17.61
18.85
-26.88
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-18.41
23.43
26.19
13.79
15.08
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14.18
13.28
14.32
13.84
17.56
15.40
12.89
12.28
-12.69
3.87
7.91
2.89
29.34
-26.68
-4.17
36.08
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-37.19
-31.44
-28.32
28.59
31.18
33.30
47.09
-48.27
-32.65
57.76
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-59.23
14*26
13.84
1.59
11.52
-11.65
36.92
27.16
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31.51
-26.51
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-30.83
-26.35
3.70
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-61*82
-:23.05
1.94
56.60
21.60
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12.51
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18.+4
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-16.57
22.97
13.44
14.80
14.49
14.12
12.54
-13.63
5.51
4.16
0.83
3.03
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-1.25
4.04
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-4.20
-4.72
-3.74
3.46
4.87
4.69
4.47
-4.34
-4.54
4.50
-1.97
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4.24
2.10
-0.75
5.25
-4.42
4.85
4.87
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1.79
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-0.10
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0.66
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4.01
3.86
0.50
2.86
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-0.93
4.18
-4.24
-4.50
-4.82
-3.53
3.34
4.81
4.82
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3.85
1.84
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4.20
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4*57
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1.71
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0.68
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288
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13.28
6.04
6.58
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7.66
24.91
26.38
9.53
9.21
24.44
8.90
9.91
24.69
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24.78
9.52
9.09
27.57
27.32
17.42
1.89
2.50
12.31
15.85
10.73
15.18
18.56
15.70
13.97
25.86
11.11
15.35
-5
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3
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5
5
4
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4
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2
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18.16
14.11
11.22
14.18
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8.64
43.50
13.64
11.44
29.40
18.77
4112.7
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Biographical Note
Charles Thompson Prewitt was born on March 3, 1933, in
Lexington, Kentucky.
Thompson Prewitt.
He is the son of John Burton and Margaret
He attended the public schools in Mt. Sterling,
Kentucky, and graduated from Mt. Sterling High School in June,
1951.
All undergraduate and graduate schooling was taken at Massachusetts
Institute of Technology, where he received an S. B. in Geology in
June, 1955, an S.M.
in Geology and Geophys ics in June, 1960, and
began work on a Ph.D. in Crystallography and Mineralogy in September,
1960.
He received a National Science Foundation travel grant to
attend the N.A. T. 0. Advanced Study Institute on "Modern Methods
of Crystal Structure Determination", held at the College of Science
and Technology, Manchester, England, August 25-September 9, 1960.
He served on active duty as a second lieutenant with the
U. S.Army from January to July, 1956, and is presently a member
of the U. S. Army Reserve in the rank of captain.
Professi onal
experience includes a number of summer or part-time jobs in
government,
industry, and university positions,
including a half-
time Research Assistantship at M. I. T. during the graduate school
years, 1956-57 to 1961-62.
He is married to the former Gretchen
Beatrice Hansen of Virginia, Minnesota.
Titles of publications:
The Crystal Structure of Cahnite, Ca
X-ray Study of Pyrographite
2
BAsO4 (OH)4 (with M. J. Buerger)
(with Otto J. Guentert)
The parameters I and 4 for equi-inclination, with application to the
single-crystal counter diffractometer
Professional societies:
Sigma Xi
American Crystallographic Association
The Geochemical Society
The Mineralogical Society of America